University Micrdfilms International 300 N. Z e e b Road Ann Arbor, Ml INFORMATION TO USERS

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1 INFORMATION TO USERS This reproduction was made from a copy o f a docum ent sent to us for microfilming. While the m ost advanced technology has been used to photograph and reproduce this docum ent, the quality o f the reproduction is heavily dependent upon the quality o f the material submitted. The following explanation o f techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or target for pages apparently lacking from the document photographed is Missing Page(s). If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure com plete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication o f either blurred copy because of m ovem ent during exposure, duplicate copy, or copyrighted materials that should n o t have been filmed. For blurred pages, a good image o f the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part o f the material being photographed, a definite m ethod o f sectioning the material has been follow ed. It is customary to begin filming at the upper left hand com er of a large sheet and to continue from left to right in equal sections with small overlaps. I f necessary, sectioning is continued again beginning below the first row and continuing on until com plete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Custom er Services Department. 5. Some pages in any docum ent may have indistinct print. In all cases the best available copy has been film ed. University Micrdfilms International 300 N. Z e e b Road Ann Arbor, Ml 48106

2 Johnson, Sandra Lee THE KURATOWSKI COVERING OF GRAPHS IN THE PROJECTIVE PLANE The Ohio State University PH.D University Microfilms International 300 N. Zeeb Road, Ann Arbor, MI 48106

3 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this docum ent have been identified here with a check mark /. 1. Glossy photographs or p ag es 2. Colored illustrations, pap er or print 3. Photographs with dark background 4. Illustrations are poor co p y 5. P ag es with black marks, not original copy_ 6. Print shows through as there is text on both sides of pag e 7. Indistinct, broken or small print on several p a g e s 8. Print exceeds margin requ irem ents 9. Tightly bound copy with print lost in spine 10. C om puter printout pages with indistinct print 11. P ag e(s) lacking when material received, and not available from school or author. 12. P ag e(s) seem to be missing in numbering only as text follows. 13. Two pages num bered. Text follows. 14. Curling and wrinkled p a g e s 15. Other University Microfilms International

4 THE KURATOWSKI COVERING OF GRAPHS IN THE PROJECTIVE PLANE DISSERTATION P re se n te d in P a r t i a l F u lf illm e n t o f th e R equirem ents f o r th e D egree D octor o f P h ilo so p h y in th e G raduate School o f The Ohio S ta te U n iv e rs ity By Sandra Lee Johnson, A.B., M.S. * * * * * The Ohio S ta te U n iv e rs ity 1982 R eading Committee* Approved By Henry H. G lover John P. Huneke G. N e il R ohertson Thomas A. Dow ling A d m ser D epartm ent o r M athem atics

5 D ed icated w ith lo v e and a p p re c ia tio n to Cleon F. O chsner.

6 ACKNOWLEDGMENTS I w ish to th an k th e many p e o p le, who h e lp e d me i n t h i s p r o j e c t. The e n t i r e l i s t would he to o long t o in c lu d e h e re, h u t a few p eo p le need t o he i d e n t i f i e d. My a d v is o r, Henry G lover, was o f im m easurable h e lp h o th d u rin g th e w r itin g o f t h i s p a p e r and in th e y e a rs b e fo re when he in tro d u c e d me to many a re a s o f Topology. Manohar Maxi an, Dan Lew is, and B i l l Hare a re a few o f th e e x c e lle n t i n s t r u c t o r s who c o n tr ib u te d to my e d u c a tio n in M athem atics. I a ls o w ish to th an k my f r ie n d s and fa m ily members, whose su p p o rt h as been in s tru m e n ta l in th e com p letio n o f t h i s p a p e r. The h e lp o f B ev erly and P a u l Johnson, M arie L in d q u is t, P a t R ie p e r, Deb W ilcox, and K athryn Jak es i s a p p re c ia te d. My s p e c ia l th an k s go to Roger R ice and Cleon O chsner.

7 VITA December 27, ^ I97I+ -I976 Born - S c o tts b lu f f, N ebraska A.B., N ebraska W esleyan U n iv e rs ity L in c o ln, N ebraska G raduate T eaching A s s o c ia te, D epartm ent o f M athem atical S c ie n c e s, Clemson U n iv e r s i t y, Clemson, South C a ro lin a 1976 M.S., Clemson U n iv e rs ity, Clemson, South C a ro lin a G raduate T eaching A s s o c ia te, D epartm ent o f M athem atics, The Ohio S ta te U n iv e rs ity, Colum bus, Ohio FIELDS OF STUDY M ajor F ie ld : M athem atics S tu d ie s in Topology. P ro fe s s o rs Graham Toomer, Henry G lover, Dan B u rg h elea and W illiam H are. S tu d ie s in A lg eb ra. P ro fe s s o rs Manohar Madan, B ostw ick Wyman and Jo h n Luedoman. S tu d ie s in C o m b in ato ric s. P ro fe s s o rs D ijen R ay-c haudhuri, A lan Sprague and J o e l B raw ley. iv

8 TABLE OF CONTENTS Page DEDICATION ACKNOWLEDGMENTS VITA LIST OF TABLES LIST OF FIGURES ii iii iv vi vii Chapter APPENDIXES 0. INTRODUCTION 1 1. SOME BASIC DEFINITIONS AND THE KURATOWSKI COVERING Definitions Kuratowski Covering SUBPOSETS OF ( l ( P ), < ) Types I and Types III and IV ( I ( P ), < ) and IN (p) 1*3 2.1* I n te r s e c tio n P r o p e rtie s 1*7 3. ( I ( P ), < ) Multiple splittings between the same graphs Types V and VI E x c e p tio n s C o n clu sio n s 68 Appendix A K uratow ski C overings f o r ( l ( P ), < ) 69 Appendix B C ross R eferen ce f o r S p l ittin g s 166 A ppendix C D iagram s 173 BIBLIOGRAPHY 179 v

9 LIST OP TABLES T able Page 1. Changes in homotopy ty p e f o r components o f K fl H f o r Types I and I, s p l i t t i n g s Changes in homotopy ty p e f o r components o f K fl H f o r Type I, I s p l i t t i n g s. 51*- 3. K uratow ski C overings f o r ( l ( P ), < ) U. C ross R eferen ce f o r S p l ittin g s 166 v i

10 LIST OF FIGURES F ig u re i. k^ 3 = s3(1^ ) K 5\ {(1, U), (2, 5)) 2. k -g rap h s 3. A1 k. V Two Kc- s 5 5- B7 * Two K3,3,S 6. B7 s and K~ «3> Al + C1 = s9 ( l, 2, 3, k, 6, 8) A1 X C2 " D1 - S7 ( 1, 3 ) C2 X { ^ 9 )) B3 * F1 = Sl ( ^ 6, 7 )B3 X f(2 ' 3)> 3)> {S> 7)> ( 5 ' Bl " D3 " S7 ( l, k) B1 X U1 ^ (5^ 6)> (2, 3)}

11 F ig u re + S9 ( i, 2 ) C^ X { (2 ' 8 ) ' <k>7)1 B5 - S2 (5, 6, 8 ) B5 X{<5' 8 >' 7 ) 3 Cl " Cl l " S5 ( l, 2, 3, M Cl X { ( ^ )} E3 ^ F5 = 2 1. K 0 H and K fl H f o r 2 2. Embedding a t r i a n g l e o p p o s ite a v a le n c y B efore Type IV S p l ittin g 2b. K in a Type IV s p l i t t i n g * D2 = S2 ( l, l t ) C3 X C (1 1 )] ^ S9 ( 1,2, 5 A X ( ( 0 9 ) ) E3 8 ' * S11 (1, 8 ) E3 8 X '< 0 11>) B1 * B2 * D7 29. Homotopy ty p e s o f components o f K 0 H 30. P o s s i b i l i t i e s f o r p.

12 F ig u re B6 - E20 = Sl ( 2, 5, *0B6 X {(2^ 3 ) ' (3 ' k) 5 )) E1 2 ^ F10 : ^ < )-E12S «i ' 5» Page 62 6h 39- y e =! S7 (6, 11) V ((5 1 0 )) 65 ho. A2 D17 Sl ( 7, 2, 3 ) A2 X {(2' 6 ) ' {3 k)> (5 ' 7 )) 66 hi. C^ F6 67 h2. D. F h3. ( I ( P ), < K) 17U kk. ( I ( P ), < ) 175 h5. A s u b s e t o f g e n e ra to rs o f ( l ( P ), < c ) 176 h6. (ln (p ), < K ) 177 hi. (IN (P ), < c ) 178

13 CHAPTER 0 INTRODUCTION The p a r t i a l l y o rd ere d s e t o f irre d u c ib le g raphs on th e r e a l p r o je c tiv e p la n e s tu d ie d in t h i s p a p e r h a s i t s o r ig in i n th e f o u r- c o lo r c o n je c tu re. The f i r s t w r itte n re fe re n c e to t h i s c o n je c tu re ap p ears in a l e t t e r from A. DeMorgan to W. H am ilton in The q u e s tio n was l e f t u n t i l 1879 a s som ething t a c i t l y b e lie v e d to be t r u e. In t h a t y e a r A. C ayley p o in te d o u t why i t was n o t c le a r t h a t fo u r o r even a f i n i t e number o f c o lo rs s u f f i c e. In th e same y e a r A. Kempe p u b lish e d w hat was b e lie v e d to be a p ro o f o f th e f o u r - c o lo r c o n je c tu r e. In 1890 P. Heawood p o in te d o u t th e e r r o r in Kempe's argum ent and proved t h a t no more th a n f iv e c o lo rs a re needed to c o lo r any map on th e sp h e re. Heawood ex ten d ed th e s u b je c t when he looked a t map c o lo rin g problem s f o r o th e r s u rfa c e s. 'T hese a re sp h e re s w ith h a n d le s o r c ro s s -c a p s a tta c h e d. A sp h ere w ith p h a n d le s w i l l be denoted 3^, w h ile a sp h ere w ith q c ro s s -c a p s w i l l be den o ted. Heawood showed th a t x(s p ) < 7 + V l + W p f o r P > 1, 1

14 where x(s ) = n i f ev e ry map on S i s c o lo ra b le w ith n o r x ' Jr few er c o lo r s and th e r e i s a map which i s n o t n - 1 c o lo r a b le. E q u a lity was shown by G. R in g e l and J.W.T. Youngs in The c o rre sp o n d in g theorem f o r n o n -o rie n ta b le s u rfa c e s was proved in 195^ t>y G. R in g e l. X(Sq ) = 7 + ^ l g+ 2^q, q^2, q > 1 and x ( s 2 ) = 6. These r e s u l t s were o b ta in e d by f in d in g th e s u rfa c e w ith few est h a n d le s o r c ro s s -c a p s on which th e com plete g rap h,, can be embedded. T his number i s c a lle d th e genus o f K, and i s denoted n Y(K ) f o r th e o r ie n ta b le c a s e, S, and Y(K ) f o r th e non- ' n P n o r ie n ta b le c a se, Sq. The e q u a tio n s Y(Kn ) = -(.n- : -3] (n j # n > 3, V(Kn ) = ( n - 3.?5(n - ^ ). j, n / T, n > 3, and Y(K? ) = 3. w ere th e r e s u l t s needed to p ro v e "Heawood's C o n je c tu re ", ( c. f. R in g e l [R ]). Two obvious q u e s tio n s a r i s e from t h i s work. The f i r s t q u e s tio n i s ; g iv en a g raph G, w hat i s i t s genus? The second q u e s tio n i s :

15 g iv en a s u rfa c e M, can a l l g rap h s which embed in M he c h a r a c te r 3 ized? T his d i s s e r t a t i o n d e a ls w ith in fo rm a tio n needed t o lo o k a t th e l a t t e r q u e s tio n. S p e c if ic a lly th e s u rfa c e S.^, th e r e a l p ro j e c t i v e p la n e, i s c o n s id e re d. In 1930 K. K uratow ski answ ered t h i s q u e s tio n co m p letely f o r th e sp h e re. He found t h a t KR and K a re th e only fo rb id d e n to p o lo g ic a l subgraphs f o r a graph t o he p la n a r, i. e. G i s p la n a r i f and o n ly i f G c o n ta in s no subgraph homeomorphic to K^ or K. These graphs are thus characterized as 5 3,3 b e in g th e homeomorphy ty p e s o f graphs which do n o t embed in th e sp h e re, b u t such t h a t any p ro p e r subgraph w i l l embed in th e sp h e re, i. e. irre d u c ib le g ra p h s. I n 1980 D. A rchdeacon [A] showed t h a t th e s e t o f 103 graphs given by H. G lover, P. Huneke, and C.S. Wang [GH W^] i s a com plete l i s t o f th e irre d u c ib le graphs f o r th e r e a l p r o je c tiv e p la n e. T his s e t i s th e s u b je c t o f t h i s d i s s e r t a t i o n. We s h a ll fo cu s on two id e a s o f i n t e r e s t co n c ern in g th e s e t, l(m ), o f i r r e d u c ib le graphs f o r a s u rfa c e M. The f i r s t i s a n a tu r a l p a r t i a l l y o rd e re d s e t, (l(m ), < ), g e n e ra te d by a s p l i t t in g and d e le tin g o p e ra tio n ( c. f. [GHW^]). The second i s a conj e c tu r e about th e s tr u c tu r e o f th e s e ir r e d u c ib le g ra p h s. n G I(M ) l e t k(g) = min [ n : G = U H. and H. i s homeomorphic i = to Kr o r K, _ f o r each i }. The K uratow ski c o v e rin g c o n je c tu re P 3} j ( c. f. [A ]) i s : F or k(g) < 2p + l f o r G l(s p ) and k(g) < p + 1 f o r G l ( S ).

16 T his c o n je c tu re was known to he tr u e f o r, a s s ta te d by A rchdeacon [A] b u t i s e x h ib ite d f o r th e f i r s t tim e h e re. T his d i s s e r t a t i o n p r e s e n ts a co n n e ctio n betw een th e se two s tr u c tu r e s, th e p a r t i a l l y o rd ere d s e t f o r l ( S 1 ) and th e K uratow ski c o v e rin g s f o r th e s e g ra p h s. In p a r t i c u l a r we p ro v e th a t f o r ev ery graph X 1 ( 1 ^ ) \ {I>2, D^) th e r e i s a graph Z 1 ( 8 ^ ) \ {DD^} such t h a t Z < X and Z < where i s a p a r t i c u l a r m axim al graph 2 U K and such th a t a l l th e s p l i t t i n g and d e le tin g o p e ra tio n s K3 in v o lv e d a re o f two ty p e s which can be d e s c rib e d i n term s o f a K uratow ski co v e rin g f o r X in i(s ^ ). The f i r s t ty p e o f o p e ra tio n ex ten d s th e n o tio n o f changing K^ to 3 as ^ T^ e second su b d iv id e s a t l e a s t one edge in th e K uratow ski su bgraphs, (se e Theorem 2.1 ). T h is connects B^ w ith a l l b u t two o f th e o th e r ir r e d u c ib le graphs f o r th e r e a l p r o je c tiv e p lan e th ro u g h s p l i t t i n g s which a re n a tu r a l e x te n s io n s o f th e s p l i t t i n g f o r th e p la n e, K^ changing t o K, - as in F ig u re 1. 0, J A g e n e r a liz a tio n o f t h i s r e s u l t would prove th e K uratow ski co v e rin g c o n je c tu re w hich in tu r n m ight le a d to s e v e r a l o th e r r e s u l t s : l ) A new p ro o f o f Heawood1s c o n je c tu r e. 2) The comput a t i o n o f th e genus o f o th e r r e g u la r g rap h s and hence th e genus o f groups ( c. f. [Wh] ). 3) Computer g e n e ra tio n o f l( S ) and l ( S ) s s g + 1 ^ 2g + l knowing th e s in g le elem ent U K3 Kk l ( S ) 5 g and U K3 K,_ l( S ). 5 g

17 In c h a p te r 1, th e n e c e s s a ry background m a te r ia l i s fo llo w e d by a d e s c r ip tio n o f th e p o s s ib le K uratow ski c o v e rin g s f o r an ir r e d u c ib l e g ra p h. A fo u n d a tio n f o r th e c l a s s i f i c a t i o n o f th e s p l i t t i n g and d e le tin g o p e ra tio n s i s l a i d by exam ining th e e f f e c t s o f a s p l i t t i n g on a K uratow ski subgraph o f th e irre d u c ib le g rap h. C hapter 2 g iv e s a c l a s s i f i c a t i o n o f th e s p l i t t i n g s i n ( i( S ^ ), < w hich can be d e s c rib e d in term s o f a K uratow ski c o v e rin g. T h is c l a s s i f i c a t i o n i s u se d t o g e n e ra te s e v e r a l p a r t i a l l y o rd ere d s u b s e ts. The main r e s u l t s o f t h i s d i s s e r t a t i o n concern th e components and maximal elem en ts o f th e s e s u b p o s e ts. The su b p o set ( l ( S, ), <v ) 1 JV u s in g a l l o f th e "good" s p l i t t i n g s, h as i n t e r e s t b ecau se o f i t s two com ponents and s m a ll number o f m axim al elem en ts (se e Theorem 2.2 ). ( l ( S n ), < v ) c o n s id e rs th e ty p e s o f s p l i t t i n g s w hich can be de- 1 " K1 s c rib e d by to p o lo g ic a l m ethods (se e Theorem 2.1 ) and ( i( S ^ ), < ) c o n s id e rs a r e s t r i c t i o n o f th e K uratow ski c o v e rin g s allow ed (see Theorem 2.3 ). F in a lly in theorem s 2.h and 2.5 l ( S 1 ) i s r e s t r i c t e d to th e s e t o f i r r e d u c ib le graphs w hich do n o t c o n ta in d i s j o i n t k -g ra p h s. Some p r o p e r tie s o f th e i n te r s e c tio n o f th e K uratow ski c o v e rin g com plete th e c h a p te r. The l a s t c h a p te r d iv id e s th e s p l i t t in g s whose d e le tio n s can n o t be e x p la in e d in term s o f a K uratow ski c o v e rin g i n to two main c la s s e s. The m inim al number o f ty p e s in d i c a te s a c o n n e c tio n betw een th e K uratow ski c o v e rin g s and th e e n tir e p o s e t ( i ( S ^ ), < ), however th e la c k o f com plete knowledge about th e K uratow ski c o v e rin g o f th e r e s u l tin g graph makes th e s e c la s s e s l e s s u s e f u l f o r g e n e ra l th eo rem s. Appendix A g iv e s a com plete l i s t

18 o f a "good" K uratow ski c o v e rin g f o r each s p l i t t i n g in th e g e n e ra tin g s e t o f ( 1 ( 8 ^, < ). A ppendix B i s a c ro s s - re f e r e n c e f o r Appendix A and Appendix C h as th e r e le v a n t diagram s.

19 CHAPTER 1 SOME BASIC DEFINITIONS AND THE KURATOWSKI COVERING S e c tio n 1.1 D e f in itio n s I t i s assumed th a t th e re a d e r i s f a m ilia r w ith th e b a s ic d e f i n i tio n s o f graph th e o ry and to p o lo g y. I f n o t, see W hite [Wh]. The fo llo w in g a re th e d e f i n i tio n s needed in t h i s d i s s e r t a t i o n. D e f in itio n : A f i n i t e graph G i s s a id t o embed in a s u rfa c e M, G c M, i f some geo m etric r e a l i z a t i o n o f G i s homeomorphic to a subspace o f M. D e f in itio n : A graph G w ith o u t is o la te d v e r t i c e s i s c a lle d an irre d u c ib le g rap h f o r a s u rfa c e M i f G does n o t embed in M, b u t g\ e does embed in M f o r any edge e o f G. Given a s u rfa c e M, l(m ) w i l l denote th e s e t o f homeomorphy c la s s e s o f irre d u c ib le g rap h s f o r M. Theorem (K uratow ski) fk ]: I ( Sq) = 3 ), where d e n o te s th e sp h e re. The two g ra p h s, Kc and K-, w i l l be c a lle d th e K uratow ski g rap h s in th e rem ain d er o f t h i s p a p e r. For a com plete graph on f iv e v e r t i c e s, K ^, w ith v e rte x s e t { v ^ vg, v ^, v ^, v ^ ) th e

20 notation is used. If the three, three partition of the vertex set of a complete bipartite graph, L is (v, v0, Ji j X 2 v^) and (v^, v,_, }, then the notation is used. This notation is extended when needed for a subgraph of a given graph with valency two vertices. For example it is possible for G to contain a subgraph K which is homeomorphic to and has vertex set {v^, v^, v^, v^, v,., v^) with v^ of valency two in K, adjacent to v and v<_, the other edges as necessary for K to be homeomorphic to. The notation in this case is Definition: The real projective plane, denoted P or S1, 2 is the space obtained from the two-sphere, S, by identifying 2 each point X of S with its antipodal point - X. When the surface under consideration is the projective plane the following terms will be used. A graph which embeds in P is called projective, while a graph which does not embed in P is called non-projective. A graph which is non-projective and not irreducible will be called reducible. Theorem (Glover, Huneke, and Wang) [GHW^] : l(p) contains at least 103 graphs.

21 Theorem (A rchdeacon) TA] : l( P ) c o n ta in s e x a c tly 103 g ra p h s. T his d i s s e r t a t i o n i s a stu d y o f th e s e 103 g ra p h s. I n o rd e r to d is c u s s them, each h as been named, as in [GHW^] w ith a p o s itiv e in te g e r s u b s c r ip tin g a c a p i t a l l e t t e r. The graphs named w ith th e same l e t t e r have th e same B e t ti number, e - v + k, where th e graph h a s e edges, v v e r t i c e s and k com ponents. The numbers i n d i c a te a le x ic o g ra p h ic a l o rd e rin g o f th e v a le n c y sequence. Definition: Let v be a vertex of a graph G with v adjacent to the vertices x,,...x., x,... x. A new graph 1 ' j 0+1 n S / v G is given as follows: delete v and all edges xj / incident with v from G. S,» G is the remaining graph V\ X2_^ x j / plus the following additional elements, v' and v" are the new vertices, e is an edge between v' and v", f(v', x ^) 1 < i < j ) and {(v", x^) j + 1 < i < n ) a re th e new ed g es. Let S G denote one of the graphs S / v G for VCx^... x^.) some s e le c tio n x.,...x.. 1" 0 P ro p o s itio n 1.1 : I f v i s a v e rte x o f th e graph G a d ja c e n t to e x a c tly n v e r t i c e s, x,, x,...x, th e n S,»G and J V 2 n "nx^) S / \ G a re homeomorphic to G. n x j,., xn - 1 ; P ro o f: S / \ G and S, v G a re b o th s u b d iv is io n s v (x 1 ) v(xv... xn _1 ) o f G and h en ce homeomorphic t o G.

22 Definition; 10 An S-operation,called a splitting of a vertex in G, is the process of producing an S G, in which it is assumed that S G = S /» G, 2 < j < n - 2. v v ^,... Xj) ' In all the examples where the vertices of a graph G are labeled with positive integers, the vertices of S G will inherit the same labels, as corresponding vertices of G except 0 and the label of v will be used for v' and v", the two new vertices in the definition of Sv G. Notice that G can be obtained from S^G by contracting e the edge between v' and v" so that they become a single vertex. Proposition 1.2; If G <{: M, then G ^ M. For the proof see Lemma 1.1 of [GHW^]. As in [GHW^] this proposition implies; Corollary 1.1; If G l(m) then a subgraph of each G is homeomorphic to a graph in l(m). Definition; For X, Z l(m), X < gr) Z means there is a vertex, v V(z) and a set of edges Y c E(Sv Z) for some choice of Z such that X is homeomorphic to S^zX y. Definition; An SD - operation, called splitting and deleting for a graph Z, is the process of producing X ^ S^zX y in l(m), given Z l(m) and the S-operation which results in S^Z. Definition; (l(m), < ) is the reflexive transitive relation on l(m) generated by successive SD - operations, i.e. H < K if and

23 o nly i f th e re e x is t s a sequence (IL 0 < i < n, and IL l(m )} where H w, K» H and H. < H.,, f o r i from 0 to O n x - SD l + l n - 1. P ro p o s itio n 1.3 : (l(m ), < ) i s a p a r t i a l o r d e r. F or th e p ro o f see Lemma 1.2 [GHW^]. F or example ( l( S n ), < ) c o n s is ts o f ( (K- _, K,.), K < K, J y IP J) as in F ig u re 1. The g e n e ra tin g s e t, (l(p), <sp)> (*(P)> <) i s g iv en in th e a p p e n d ic e s. I t i s in ta b le form in A ppendices A and B and in a d iag ra m i n A ppendix C.!3(1, *0K5X ((1' k)> (2' 5)} The following definitions are fundamental to the study of the relationship between (l(p), < ) and the Kuratowski covering conjecture. Definition: A poset (S, ^ ) is connected if for every X, Y S there exists a sequence 0 < i < n, Z^ S} where X = ZQ, Y = Zn, and Z^ and + 8X6 comparable, i.e. Z. h Z.. or Z.. ^ Z., for i between 0 and n-1. X X + X X + X X

24 D e f in itio n : A component o f a p o s e t (S, ) i s a connected 12 su b p o set (K, ^ ) such t h a t i f (L, \* ) i s a co n n ected su b p o set o f (S, ^ ) which c o n ta in s (K, ) th e n (K, ^ ) = (L, ^» ). D e f in itio n : F or a p a r t i a l l y o rd ere d s e t (S, ^ th e s e t M ^ S i s th e s e t o f maximal elem en ts o f (S, 4 ), i.e. x M S i f and o n ly i f f o r e v ery z S such t h a t x ^ z we have x = z. D e f in itio n [GH^ : A subgraph A o f a g raph G i s c a lle d a k -g rap h i f th e r e e x i s t s a g raph B such t h a t A c B c G and e i t h e r ( i ) A i s homeomorphic t o K0 - and B i s homeomorphic to c.)5 K» _ w ith one o f th e cu b ic v e r t i c e s o f B n o t in A, J? 3 (ii) A i s homeomorphic t o and B i s homeomorphic to w ith one o f th e v a le n c y fo u r v e r t i c e s o f B n o t in A, o r (iii) A i s homeomorphic t o and B i s homeomorphic to S K,. w ith th e cu b ic v e r t i c e s o f B n o t in A. v 5 In F ig u re 2, th e k -g ra p h s have s o lid edges and th e subgraphs o f G w hich c o n ta in them in c lu d e a ls o th e dashed e d g es. Figure 2 k-graphs

25 13 D e f in itio n ; K, H i s a K uratow ski c o v e rin g o f th e graph G i f K and H a re subgraphs o f G, G = K U H, and K and H a re each homeomorphic to one o f o r ^. S e c tio n 1.2 K uratow ski C overing Each member o f l(p) h a s a K uratow ski c o v e rin g as was n o te d by A rchdeacon [A]. The diagram s in th e appendix o f h i s d i s s e r t a t i o n [A] em phasize two k -g ra p h s. The s ix g rap h s D^, D^Q, D ^, and have K^' s w hich a re c o n ta in e d in K^ and hence a re n o t u s e f u l in f in d in g a K uratow ski c o v e rin g. I n th e o th e r graphs a Kurato w sk i c o v e rin g can be re c o v e re d from A rchdeacon's appendix. There a re a few graphs which have a unique K uratow ski c o v e rin g, see F ig u re 3, however m ost o f th e g rap h s in l(p) have s e v e r a l c h o ic e s f o r c o v e rs. F ig u re 3 A.

26 Not o n ly a re th e r e c h o ic e s o f v e rte x s e t s, a r c s betw een v e r t i c e s, and c h o ic e s b ecau se o f symmetry b u t th e r e a re o c c a s io n a lly c h o ic e s o f w hich p a i r o f K uratow ski graphs to u s e. F o r example F ig u re s b, 5, and 6 g iv e B as a u n io n o f two Kc's, two K0, s and one * 7 5 3,3 o f each, r e s p e c tiv e ly. The e f f e c t o f an S -o p e ra tio n on any K uratow ski subgraph o f a graph G i s lim ite d to th re e p o s s i b i l i t i e s. P r o p o s itio n 1.4 ; L et K be a K uratow ski subgraph o f G l(p) and l e t v K. The e f f e c t o f an S -o p e ra tio n on K i s : (51) A ll edges in K th a t a re in c id e n t w ith v in G a re in c id e n t w ith one o f v ' o r v" in S^G, i.e. K i s unchanged, o r (52) E x a c tly one edge o f K t h a t i s in c id e n t w ith v in G i s in c id e n t w ith v ' (o r w ith v ") in S^G, i.e. An edge o f K i s su b d iv id e d, o r (53) v i s in c id e n t w ith fo u r edges o f K and v 1 and v" a re each in c id e n t w ith two o f th e s e in S^G.

27 P ro o f: Any v e rte x in K h a s v a le n c y 2, 3, o r U s in c e K i s homeomorphic t o ^ o r 15 i ) I f v h a s v a le n c y two, th e n th e p o s s ib le p a r t i t i o n s a re 0, 2 and 1, 1. These a re p o s s i b i l i t i e s SI and S2, r e s p e c tiv e ly. i i ) I f v h a s v a le n c y t h r e e, th e n th e p o s s ib le p a r t i t i o n s a re 0, 3 and 1, 2. These a re c a se s S I and S2, r e s p e c tiv e ly. iii) I f v h as v a le n c y f o u r, th e n th e p o s s ib le p a r t i t i o n s a re 0, k, 1, 3, and 2, 2. These a re c a se s S I, S2, and S3, r e s p e c t i v e l y. In c o n s id e rin g a K uratow ski subgraph K o f a graph G, an a rc from v to v ' can be used t o r e p re s e n t th e edge (v, v ' ) in K. The fo llo w in g d e f i n i tio n g iv e s a n o ta tio n f o r th e edges in th e s e a r c s. D e f in itio n : Given K a K uratow ski subgraph o f G, v, y, two v e r tic e s o f v a le n c y g r e a te r th an two in K and Kp th e arc v *y (v, xx ), (Xp, x )... (xn _1, xn ) in K where *n = y, th e x^'s a re d i s t i n c t, n > 1, and each x ^, i < n i s o f v a le n c y two in K. Let K P denote th e s e t o f edges in K ^ v,y Given K, a K uratow ski subgraph o f G, and an S -o p e ra tio n, th e n t h i s S -o p e ra tio n r e s t r i c t e d t o K r e s u l t s in a grap h, denoted S^K, w hich as im p lie d b y P r o p o s itio n l. 1* c o n ta in s a K uratow ski subgraph, c a l l i t S^K.

28 D e f in itio n : i ) S K = S K = K i f v ji K, o r > v v ' i i ) S^K = S^K in c a se s SI and S2 o f p r o p o s itio n 1.4, o r iii) S K = S k\(kp U KP ) in c a se S3 o f P ropo- V V X.^, X g X ^ J s i t i o n 1.4 where v ' i s in c id e n t w ith an edge o f K and v, x1 I t i s c l e a r th a t f o r G, th e S -o p e ra tio n w hich r e s u l t s in S^G and any K uratow ski c o v e rin g G = K U H we have S^G = S^K U S^H U (e), I n th e r e s t o f t h i s p a p e r we w i l l c o n s id e r th e c ases where th e r e s u l t o f an S D -o p e ra tio n i s th e u n ion o f two K uratow ski subgraphs w hich come from some K uratow ski c o v e rin g o f th e o r ig i n a l g rap h, i.e. for G = K U H we get S^gXy = S^K U S^H. I t i s im p o rta n t to n o te th a t S^gXy i s known and th e K uratow ski co v e rin g i s so u g h t.

29 CHAPTER 2 SUBPOSETS OF ( l ( P ), < ) T h is c h a p te r c l a s s i f i e s th e f o u r ty p es o f S D -o p e ra tio n s from G t o S g\ Y where th e r e i s a K uratow ski c o v e rin g o f G = K U H such t h a t S ^ g X y = S^K U S^H. The su b p o sets o f ( l ( P ), < ) w hich a re g e n e ra te d by th e s e "good" S D -o p e ra tio n s a re exam ined. The s e t l( P ) i s th e n r e s t r i c t e d to th e 52 graphs w hich do n o t 2 c o n ta in d i s j o i n t k -g ra p h s. Here th e s p e c ia l graph B.. = U K i s K3 5 n o t o n ly connected to e v e ry graph b u t th e r e i s a t m ost one r e v e r s a l o f o rd e r, i. e. f o r G in l ( p ) w hich does n o t have d i s j o i n t k -g ra p h s, th e r e i s a n o th e r such grap h H such t h a t H < B^ and H < G, where th e p a r t i a l o rd e r i s r e s t r i c t e d to c e r t a i n ty p e s o f SD- o p e ra tio n s. A d is c u s s io n o f th e changes in th e i n te r s e c tio n o f th e two K uratow ski subgraphs d u rin g an S D -o p e ra tio n com pletes th e c h a p te r. S e c tio n 2.1 Types I and The m o tiv a tio n f o r Type I s p l i t t i n g s i s th e n a tu r a l s p l i t t i n g s o f a K^ in to K^ ^ as in F ig u re 1. 17

30 D e f in itio n ; An S D -o p e ra tio n changing G t o S ^ gvy in th e 18 g e n e ra tin g s e t o f ( l ( P ), < ) i s c a lle d Type I i f th e r e i s a K urato w sk i c o v e rin g, K U H = G, which s a t i s f i e s th e fo llo w in g c o n d i t io n s ; i ) K h as v a le n c y fo u r v e r t i c e s v, x ^, X y x^. i i ) I f e. i s th e edge in K in c id e n t w ith v, th e n 1 v, x. 1 in S G e, and e a re in c id e n t w ith v ' and e 0 and e. a re v A in c id e n t w ith v". i i i ) v i s n o t a v e r te x in H, o r th e e f f e c t o f th e S -o p e ra tio n on H i s case S I o f P ro p o s itio n 1.^, i. e. S^H = H. iv ) Y = K P U KP \E(SH). Xl> 2 3 *» I f an S D -o p e ra tio n changing G to S ^ g X y i s o f Type I, where G = L U M i s a K uratow ski c o v e rin g w hich f i t s th e r e q u ir e m ents f o r a Type I s p l i t t i n g th en Type I X, X = L o r M, w i l l he u sed to in d ic a te which subgraph, X, s a t i s f i e s i ) and i i ) in th e d e f i n i tio n o f Type I. N ote t h i s says X i s homeomorphic to K^ i n G and S^X i s hom eom orphic t o K^ ^ D e f in itio n : The edges in K P U K P _ a re s a id to V 2 3 ^ be e lim in a te d from K. D e f in itio n ; An S D -o p e ra tio n changing G to S ^ g X y in th e g e n e ra tin g s e t o f ( l ( P ), < ) i s c a lle d Type I, I i f th e r e e x i s t s a K uratow ski c o v e rin g, G = K U H, w hich s a t i s f i e s th e fo llo w in g

31 c o n d itio n s ; 19 i ) K h as v a le n c y fo u r v e r t i c e s v, x ^, x x ^, x^. H has v a le n c y fo u r v e r t i c e s v, y ^ y^, y y y^ where i f y i {x1, x^, Xy x^} th en y,. = x±. i i ) I f e. i s th e.e d g e in K in c id e n t w ith v and v, x. e. i s th e edge in IL in c id e n t w ith v, th en in S G V' y i A A J 4. 4.U. J A A e^, e^, e y e^ a re in c id e n t w ith v 1 and e ^, e ^, a re in c id e n t w ith v". i i i ) Y = K P U KP U HP U HP \ 1* 2 3 ' ^ yl* y2 y3 yk E(S K U SMB). ' v v Examples o f Type I and Type I, I s p l i t t i n g s i l l u s t r a t e th e d e f i n i tio n s. The s im p le s t case i s A1 ->. Given th e la b e lin g in F ig u re 7, A ^ s K uratow ski c o v e rin g i s, H = ^ 7* ^ 9} and A1 -> i s a Type I H s p l i t t i n g, where 2 3 ^ 6 8) Al ^ ((5> 7 ), (6, 8 ) ). Since ( 5, 7) and ( 6, 8) a re e lim in a te d from H and a re n o t in S K, th e y a re d e le te d from th e graph S / _. y ^9 $9 ^9 6, 8)Ai *

32 20 \ / I \ X s \ I / I n W\ F ig u re 7 C1 s 9 ( l, 2, 3, U,6, 8 ) A1X «5 7 ), <6 8 >> -» i s an example where an edge i s e lim in a te d from th e b u t n o t d e le te d from th e g rap h. W ith th e fo llo w in g diagram and la b e l in g f o r C2, th e K uratow ski cover i s K =! g ^ r ) i l l :) and 3 ) ^ 2 ^ R e s tr ic ^ed to K s p l i t t i n g is S ^ ^ k \ {(8, 9 ), (3, 6 ), (6, l ) ), however ( 3, 6 ) and ( 6,1 ) a re b o th used in H=S^^ and hence a re n o t d e le te d from S /, C. As seen in F ig u re 8, D i s th e u nion o f 7 \ 1 * 5 ) S7(l, 3)K = \ / 803 S7(l, 3)H = ~ \6 (\ 63 h k5 20 /

33 21 F ig u re 8 C2 * D1 = S7 ( 1, 3 ) C2 M ( 8 9 )) An example o f a Type I, I s p l i t t i n g i s -*. The K ura- / 6 1>3\ / 2 1. \ tow ski c o v e rin g f o r F ig u re 9 i s K = g I, H = I ^ y The s p l i t t i n g F^ = ^ ^ B 3\ ( ( 2, 3 ), (*+, 5 ), ( 6, 7 ), (5, 8)} can he r e s t r i c t e d to K and H in th e obvious way. S ince none o f th e edges e lim in a te d from K and H w ere in E(S^K U S^H) th e y a re a l l in Y. 4 / F ig u re 9 B3 -v g B3V ( 2, 3 ), (k, 5), (6, l ), (5, 8)}

34 A f i n a l example o f a Type I, I s p l i t t i n g i s -* Eg. T h is i s 22 a Type I, 1 s p l i t t i n g where an edge o f th e i n te r s e c tio n, K H H, i s in Y. U sing th e s p l i t t i n g = S ^ {(1, 4 ), (5, 6 ), (2, 3 )) when r e s t r i c t e d to e i t h e r K o r H r e q u ir e s t h a t ( l, U) he e lim in a te d from h o th K and H, t h i s means t h a t ( l, 4) m ust be in Y. F ig u re 10 g iv e s th e la b e lin g as w e ll as a v is u a l r e p r e s e n ta tio n o f t h i s s p l i t t i n g. F ig u re 10 B - Eg = S7 (1 ^ ) B1 \ {(1, 10, (5, 6 ), (2, 3 )) D e f in itio n : An S D -o p e ra tio n changing G to g\ Y in th e g e n e ra tin g s e t o f ( l ( P ), < ) i s c a lle d Type i f th e r e i s a Kuratow ski c o v e rin g G = K U H which s a t i s f i e s th e fo llo w in g c o n d itio n s : i ) The e f f e c t o f th e S -o p e ra tio n on K i s case S2 o f Propos i t i o n l A, i. e. an edge o f K i s su b d iv id e d. i i ) v i s n o t a v e r te x o f H o r th e e f f e c t o f th e S -o p e ra tio n on H i s case S I o f P ro p o s itio n 1.^, i. e. H i s n o t changed. i i i ) Y = 0.

35 Given an S D -o p e ra tin g changing G to Sv G \ Y w hich i s o f 23 Type and a K uratow ski c o v e rin g G = L U M, which s a t i s f i e s th e req u irem e n ts f o r a Type s p l i t t i n g th e n Type X, X = L o r M, w i l l be u sed to in d ic a te which K uratow ski subgraph, X, s a t i s f i e s i ) in th e d e f i n i tio n o f Type s p l i t t i n g s. D e f in itio n : An S D -o p e ra tio n changing G to g\ Y, in th e g e n e ra tin g s e t o f ( l ( p ), < ) i s c a lle d Type, i f th e r e i s a K uratow ski c o v e rin g, G = H U K, such t h a t : i ) The e f f e c t o f th e S -o p e ra tio n on b o th K and H i s case S2 in P ro p o s itio n l A, i. e. each has an edge su b d iv id e d. i i ) Y = f. D e f in itio n : An S D -o p e ra tio n changing G to S ^ g X y in th e g e n e ra tin g s e t o f ( l ( p ), < ) i s c a lle d Type I, i f th e r e e x i s t s a K uratow ski c o v e rin g G = K U H, such t h a t ; i ) K s a t i s f i e s i ) and i i ) in th e d e f i n i tio n o f Type I. i i ) H s a t i s f i e s i ) in th e d e f i n i tio n o f Type. i i i ) Y = KP U KP \ E(S^H ). l* 2 3* k Given a Type I, S D -o p e ra tio n changing G to Sv G \ y where G = L U M i s a K uratow ski c o v e rin g th e n Type I X, Y, X = L and Y = M o r X = M and Y = L, w i l l be used to in d ic a te which subg rap h, X, s a t i s f i e s i ) o f th e d e f i n i t i o n f o r Type I, and which subgraph, Y, s a t i s f i e s i i ) in th e same d e f i n i tio n. In r e f e r i n g to th e s e c l a s s i f i c a t i o n s in g e n e ra l Type One w i l l be used to in d ic a te

36 Type I, Type I, I o r Type I, w h ile Type Two w i l l be used to i n d i c a te Type o r Type,. Examples o f Type, Type,, and Type I, s p l i t t i n g s can be seen in F ig u re s 11, 12, and 13. The f i r s t example i s E^ -> E^ and h e re th e edge (5, 7) in H i s re p la c e d by th e a rc (0, 5 ), f 8 2 5\ (5, 7) N o tice th a t 5 i s o f v a le n c y 3 in b o th K =\ 1 ^ ^ J and H = k 8\ j I, b u t in th e two r e s u l tin g K uratow ski g rap h s 5 i s o f v a le n c y 3 in ^ 7) K, w h ile 0 i s o f v a le n c y 3 in S5(l^. T h is i s a Type H s p l i t t i n g and F ig u re 11 shows th e r e s u l t. ^---- i Y & 1 2 F ig u re 11 f E, c * E9 S5 ( i, K 7) t D,_ = ^ g iven th e la b e lin g in F ig u re 12. T his example i s o f Type, and shows th a t b o th K uratow ski graphs may have an edge su b d iv id e d. K = ( 1 2^ \ 1^ 5 3 / has th e edge ^ 3) re p la c e d

37 2 3 6 by (1, 0) and. (0, 3 ), w h ile H =1 ^ ^ g I has ( l, 2) re p la c e d 25 b y (1, 0) and ( l, 2) F ig u re 12 D5 ' Sl(2, If, 5)D3 C2 -> Eg i s a Type I, s p l i t t i n g when th e K uratow ski c o v e rin g 3 5 1' K = (\ 7 \8 9 M ), H =1 6 ^ 2 1 and th e s p l i t t i n g Eg = g 1 / 2^) C gv C?, 9 )> (3, 8 ) ) a re u se d. F ig u re 13 r e p r e s e n ts t h i s s p l i t t i n g and in d ic a te s th e ^ w hich i s ^1(2 If 7 9) ^ F ig u re 13 E8 = Sl(7,9,2,M C2Xt(7 9) (3' 8))

38 From Appendix B i t can be seen t h a t Type One and Type Two s p l i t 26 tin g s acco u n t f o r 228 o f th e 1+7^ s p l i t t i n g s in ( l ( P ), < g D ) > th e g e n e ra tin g s e t o f ( l ( p ), < ). I t i s n a tu r a l t o lo o k a t th e su b p o set o f ( l ( P ), < ) which i s g e n e ra te d by th e s e 228 S D -o p e ra tio n s. D e f in itio n : ( l ( P ), < T, ) i s th e r e f le x iv e t r a n s i t i v e r e l a t i o n 1 on l ( p ) g e n e ra te d by S D -o p e ra tio n s which a re o f Type One o r Two. P ro p o s itio n 2.1 : ( l ( p ), ) i s a p a r t i a l l y o rd ere d s e t. P ro o f: ( i( P ) j < T, ) i s r e f le x iv e and t r a n s i t i v e by d e f i n i tio n. K1 Any r e l a t i o n in < v i s a r e l a tio n in < hence < v i s a n ti- K1 _K 1 sym m etric. Theorem 2.1 : a) ( l ( P ), < ir ) h as th re e com ponents (D^}, {D ) y and I ( P ) \ { D 2, D9 ). b ) M< l( P ) = {A^ Ag, By D, D, _K 1 ^22* ^2* A5^ ' P ro o f; a) F ig u re ^5 in Appendix C g iv e s enough o f th e Type One and Two S D -o p e ra tio n s t o show t h a t l ( P ) \ {DD^} i s a connected s u b se t o f l ( p ). N e ith e r D2 n o r D^ c o n ta in a subgraph homeomorphic to so can n o t be th e source o f a Type One s p l i t t i n g. F or a l l S D -o p e ra tio n o r ig in a tin g w ith Dg o r D^, Y ^ 0 so th e y can n o t be Type Two s p l i t t i n g s. DQ i s maximal in ( l ( P ), < ) hence y ~ we have th e component (D^). Dg i s th e r e s u l t o f an S D -o p e ra tio n

39 only from and h e re Y ^ 0, so Type One s p l i t t i n g s a re th e o n ly 27 p o s s i b i l i t y. A case by case check shows th e r e i s no K uratow ski co v e rin g w hich w i l l make t h i s a Type I, Type I, I, o r Type I, s p l i t t i n g, hence {Dg} i s a component. l ( p ) \ { ( D g, D^} b e in g conn e c te d and each o f {D2 ) and b e in g components im p lie s l ( p ) \ {Dg, D^} i s a component of ( l ( P ), < K ). b ) A^, Ag, B^, B^ and D^ a re maximal in ( l ( P ), < ) hence in ( l ( p ), < K ). {Dg} is a component o f ( l ( P ), < R ) so Dg i s m aximal in ( l ( p ), < K ). By checking a l l S D -o p e ra tio n w hich r e s u l t in 1 D17, ^22* ^2* ^5 Kura'*'ows^::'- co v e rin g s f o r th e s e S D -o p e ra tio n s i t can be shown th a t th e y a re a l l maximal in ( l ( p ), ). D I t sh o u ld be n o ted h e re th a t Types One and Two have th e h ig h e s t p r i o r i t y. I f i t i s p o s s ib le f o r a s p l i t t i n g to be Type One o r Two th en i t i s so l i s t e d in th e A ppendices. The p o s e t ( l ( P ), < ) has K1 th e advantage th a t th e s e s p l i t t i n g s can be u n d e rsto o d in term s o f o th e r to p o lo g ic a l m ethods. We w i l l r e tu r n to t h i s su b p o set in S e c tio n 2.3 when we r e s t r i c t o u rse lv e s to ir r e d u c ib le graphs t h a t do n o t have d i s j o i n t k -g ra p h s. In o rd e r to c l a s s i f y th e s p l i t t i n g s in th e g e n e ra tin g s e t ( l ( P ), < S I)) o f (f(p )> < ) acco rd in g to ty p e, one m ust f in d an a p p ro p ria te K uratow ski c o v e rin g. p o in t o f view by stu d y in g a graph I t i s i n te r e s t in g to re v e rs e th e G, w ith K uratow ski co v e rin g

40 G = H U K, a s p l i t t i n g Sy G and th e s e t Y = Sy G \(S v K U Sv H). In S D -o p e ra tio n s o f Types One o r Two we have exam ples where S ^ gx y i s ir r e d u c ib le, however i t can a ls o be p r o je c tiv e o r re d u c ib le as th e K uratow ski c o v e rin g changes. D e f in itio n : Given a graph G, an S -o p e ra tio n r e s u l tin g in S G and a K uratow ski c o v e rin g G = K U H, th e e f f e c t o f th e S -o p erat io n on K and H w i l l be c a lle d Case ( i, j ) where one o f K or H i s e f f e c te d as in case S i o f P r o p o s itio n 1.1+ and th e o th e r i s a f f e c te d a s in case Sj o f P ro p o s itio n l.u. P r o p o s itio n 2.2 : Given G l ( p ), th e S -o p e ra tio n which r e s u l t s in S G, and a K uratow ski c o v e rin g, G = K U H, i f th e e f f e c t o f th e S -o p e ra tio n on K and H i s Case (2, 2) o r ( l, 2) th en l ) S^K U S^H i s n o n -p ro j e c t iv e 2) i t i s p o s s ib le t h a t S^K U S^H i s re d u c ib le. P ro o f; l ) F o r Cases ( l, 2 ) and (2,2 ) S^K = S^K and S H = I T h and {e} S s K U S H hence Y = S g X ep k U 1T~H = f. v v v v v v v r P ro p o s itio n 1.1 com pletes th e p ro o f t h a t S^K U S^H i s non- p ro j e c t iv e. 2) F ig u re lu g iv e s a la b e lin g o f. F or t h a t la b e lin g S10(7, 11) E35 and th e K uratow ski c o v e rin g

41 S10(7, l l ) E35 Since F9 i s homeomorphic to s io ( 7, as e x h ib ite d by th e mapping in d ic a te d in th e v e rte x la b e lin g o f F ig u re 15, i t i s c le a r t h a t si q (7 l l ) E35 ^ no^ i r r e d u c ib le. Note t h a t t h i s i s th e c o v e rin g used in th e S D -o p e ra tio n F - = S, - /r_ 9 10(7, 11; 35 {(1,1)-)) in A ppendix A. O ther s p l i t t i n g s l i s t e d in th e Appendix as Type V I, w i l l g iv e th e same r e d u c i b i l i t y. S p l ittin g s l i s t e d in th e Appendix as Type VI show t h a t th e e f f e c t o f Case ( l, 2) can be r e d u c ib le. F ig u re l 1^ E, F ig u re 15 F S / A 9 10(7, H ) 35

42 When one o f th e su b g rap h s, K, in th e K uratow ski c o v e rin g, 30 G = K U H, i s homeomorphic t o and K i s homeomorphic to K no in fo rm a tio n about th e e m b e d a b ility o f S K U S H can be 3,3 v v d e term in ed. P ro p o s itio n 2.3 i Given G l ( p ), an S -o p e ra tio n which r e s u l t s in Sv G, and a K uratow ski c o v e rin g, G = H U K, th e n i f th e e f f e c t o f th e S -o p e ra tio n i s Case ( l, 3 ), (2, 3) o r (3, 3 ), S^K U S^H can b e i ) r e d u c ib le, i i ) ir r e d u c ib le, o r i i i ) p r o je c tiv e. P ro o f; i ) U sing B g, th e K uratow ski c o v e rin g Kg U = Bg from Appendix A (page 76), and S ^ 2 ^ ^ Bg th e n ' s ^ ' ^ 7)^K2 U ^ 7 )^ 2 con *:;a:ijls C3, w here th e e f f e c t o f th e S -o p e ra tio n i s (1, 3). U sing A^, th e K uratow ski c o v e rin g K^,, from Appendix A (page 7 0 ) and th e S -o p e ra tio n Ij ) A1 S iv es th e e f f e c t on Kx and a s Case ( 2, 3) b u t ^ ^ u ^ a T C 5r C 7 )"l = s9 (l, h, 5, 6, 7 ) *1 ^ ^ ^ 3) } which contains th e subgraph U sing th e same graph and K uratow ski c o v e rin g, th e e f f e c t o f S9 (2, k, 5, 7) Al is Case (3 ' 3) b u t ^9 ( 2X ^ 1 U S9 ( l, 5, 6, 7) (5, 7 ), (6, 8)} c o n ta in s a subgraph homeomorphic to Fg. (See A ppendix A page 7 1 ) These th r e e exam ples

43 prove p a r t i ). 31 i i ) Any S D -o p e ra tio n o f Type One i s an example t h a t th e s e can be i r r e d u c ib le. 1 3 < i i i ) For V < 4,,.. I,, l j, a K urato w sk i c o v e rin g, and th e s p l i t t i n g 3) 7 8ra ph 2 ( l 3 ) K ^ ^ 3 ) B ^ \ {(1, 3 ), (6, 7 ) ) T his i s a subgraph o f = ^ 2 ( l 3 ) ^ 7 ^ hence p r o je c tiv e. (See F ig u re 16 and Append ix A page 86). F ig u re 16 S2 ( l ^ 3 ) B7 \ ( ( l, 3 ), ( 6, 7)3 8 2 ""' 3 ^ 1 \ The K uratow ski c o v e rin g K H =1 ^J o f and th e s p l i t t i n g S ^ ^ 2 ) ch would r e s u l t i n S C, \ 9 (1, 2 ) ^ { (2, 8 ), (4, 7 ) ) w hich i s a subgraph o f D12 = S ^ 2 ) ^ ^ and hence p r o je c tiv e. (See F ig u re 17 and Appendix A page 9&).

44 32 F ig u re 17 s 9 ( l, 2 ) C^ X {(2' 8 ) ' (U 7 ) ) / 2^ \ 3 \ ( 5*3 7 \ U sing, th e K uratow ski c o v e rin g K = l 1 g 3 y, H \ 2 6 \ J and th e s p l i t t i n g ^ ( 5 g 8 ) ^B, th e graph S2 (5 ; ^ g )K U g )H = S2 ( 5, 6, 8 ) 6), (**, 8 ), ( 1, 7 ) ) i s a subgraph o f, g^b j and hence p r o je c tiv e. (See F ig u re 19 and A ppendix A page 8l ). These a re exam ples where S^K U S^H i s p r o je c tiv e and com pletes th e p ro o f o f P ro p o s itio n 2.3. O F ig u re 18 s 2 ( 5, 6, 8) B5 X f(5, 6 ), (U, 8 ), ( 1, 7 )}

45 S e c tio n 2.2 Types I and IV 33 T his s e c tio n d e s c rib e s th e rem ain in g s p l i t t i n g s w hich can be d e s c rib e d in term s o f a K uratow ski c o v e rin g. D e f in itio n : An S D -o p e ra tio n changing G t o S ^ gx y in the g e n e ra tin g s e t o f ( l( P ), < ) i s c a lle d Type I i f th e r e i s a K uratow ski c o v e rin g G = K U H which s a t i s f i e s th e fo llo w in g : i ) A ll edges o f K t h a t a re in c id e n t w ith v in G a re in c id e n t w ith v ' in S ^ G, i. e. The e f f e c t o f th e S -o p e ra tio n on K i s S I o f P ro p o s itio n 1.4. i i ) A ll edges o f H t h a t a re in c id e n t w ith v in G sure i n c i d e n t w ith v" in S G, i. e. The e f f e c t o f th e S -o p e ra tio n on H i s S I o f P ro p o s itio n 1.4. i i i ) Y = ( ( v ', v " ) }, A Type I s p l i t t i n g o c c u rs when K fl H h as an is o la te d v e rte x. T h is o c c u rs most o f te n when a connected g rap h, i. e., s p l i t s in to a graph w ith two com ponents, i. e. (se e F ig u re 1 9 ).

46 S5 ( l, 2, 3, b) c i c F ig u re 19 However i t i s p o s s ib le th a t b o th g rap h s a re connected a s in E ^ + F^. As seen in F ig u re 20, h a s th e K uratow ski C overing K = w ith K H H th e fo llo w in g : 1) The v e rte x s e t o f K H H i s {1, 2, 3, 5, 6, 7, 8, 10, 11}. 2) The edge s e t o f K fl H i s { ( l, 1 1 ), ( l, 2), (2, 3), (2, 6), (3, b), (3, 1 1 ), (6, 7 ), (7, 8 ), (1 0, 11)}. As seen i f F ig u re 21 K H H is a v e rte x and a subgraph hom otopic t o a c y c le. S^K H S^H i s j u s t th e c y c le.

47 F ig u re 20 8 K n H f o r E ^ -* F5 K fl H f o r E F, F ig u re 21

48 A check o f Appendix B shows t h a t th e o n ly improvement t h a t can be made in ( l ( P ), < ) by adding th e Type I s p l i t t i n g s would be _ K 1 to have one l e s s maximal e lem en t, A,-. Since ( v ', v " ) can n o t be an edge e lim in a te d from a K uratow ski subgraph, Type I s p l i t t i n g s can n o t be Type One s p l i t t i n g s, Y ^ <f) im p lie s th e s e a re n o t Type Two s p l i t t i n g s e i t h e r. S ince Type I i s d i s j o i n t from Types One and Two th e y a l l have th e same p r i o r i t y. I f Types One, Two, o r I a re p o s s ib le, th e y a re l i s t e d in A ppendix A. A Type IV s p l i t t i n g i s th e m ost te c h n ic a l, in t h a t i t a r is e s from th e f a c t t h a t no i r r e d u c ib le graph can have a v a le n c y th re e v e rte x as one v e rte x o f a t r i a n g l e. For in s ta n c e in G i f v has n e ig h b o rs x ^ x ^, and x^ th e n c o n s id e r th e edge e = ( x ^ Xg). I f e E(G) and G - e embeds th e n th e r e m ust be a fa c e w ith (x ^, v) and (v, Xg) on i t s boundary, s in c e th e r e i s no way t o s e p a ra te x.^ and Xg w ith o n ly one o th e r edge. As i n F ig u re 22 e can be added to G - w ith o u t any i n t e r s e c t i o n, hence G embeds. 3 F ig u re 22 Embedding a t r i a n g l e o p p o s ite a v a le n c y 3 v e rte x P ro p o s itio n 2.^-: L e t G l ( P ), and l e t x^ and Xg be a d ja c e n t to v in G. I f x^^ and x a re a d ja c e n t a ls o, th en

49 f y x j, O f(*l' XS> } i s non p r o je c tiv e. 37 F o r a p ro o f see Lemma 3.^ [GWH^]. Type IV s p l i t t i n g s w i l l be th o se which s a t i s f y P r o p o s itio n 2.b and a re n o t Type One s p l i t t i n g s. T here a re o n ly fo u r such s p l i t t i n g s, C3 * V D2 * V D1 0 * E15' a e23 * P10 D e f in itio n : An S D -o p e ra tio n changing G to S, Ng \ X2 { (x ^ x 2 ) ) in th e g e n e ra tin g s e t o f ( l ( P ), < ) i s c a lle d Type IV i f i t i s n o t Type One and th e r e i s a K uratow ski c o v e rin g G = K U H which s a t i s f i e s th e fo llo w in g : i ) K u s e s v and x^ as v a le n c y th r e e v e r t i c e s, x g as a t l e a s t v a le n c y two and th e edges (v, x^) and (x^, a re in K. ( See F ig u re 2 3 ). F ig u re 23 K in a Type IV s p l i t t i n g i i ) The e f f e c t o f S, * G on H i s S2 o f P ro p o s itio n l.u. n x x, xg ; i i i ) (x _, x ) i s n o t i n H and H i s homeomorphic t o K. JL c. 3)3 The s tr e n g th o f th e Type IV s p l i t t i n g i s th a t S / \ g \ v ^, Xgj { (x ^ x 2 ) ) = Sv H U K where K i s (S^K U { (v 1, xg ) )) \ { ( x x, x g ) }.

50 T h is i s a ^ w ith th e changes as in F ig u re V* F ig u re 2^ K in a Type IV s p l i t t i n g F or example -* Dg i s shown in F ig u re 25. W ith th e g iv en h \ la b e lin g K = 9-^ g ^ ** = ^ 7 8 f ^ ^ l ^ sp l t t t i n g D2 S2 ( l ( l4. ) C3 >S' fc1# it ) ) r e s u l t s in K = 1 ^ lfg 3 J and N o tice t h a t h e re v = 2, v ' = 2, r" Oj x^ = 1 and = k. I F ig u re 25 3 *

51 39 These seven ty p e s o f s p l i t t i n g s have th e p ro p e rty t h a t g iv en th e s p l i t t i n g and th e K uratow ski c o v e rin g o f one o f th e graphs in v o lv ed th e K uratow ski c o v e rin g o f th e o th e r i s re c o v e ra b le. These 2bj s p l i t tin g s g e n e ra te a p a r t i a l l y o rd ere d s e t. D e f in itio n : ( l ( P ), < ) i s th e r e f le x iv e t r a n s i t i v e r e l a tio n ' K on l( P ) g e n e ra te d by S D -o p e ra tio n s o f Types One, Two, I and IV. P ro p o s itio n 2.9 ; (l(p )> ) as a P ^ 'k i ^ l y o rd ere d s e t. P ro o f: By d e f i n i tio n ( l ( P ), < ) i s r e f le x iv e and t r a n s i t i v e and < i s a s u b r e la tio n o f <, hence a n tisy m m e tric. K Theorem 2.2 : a ) The components o f ( l ( P ), < ) a re (D^} and is. y I ( P ) \ [D9 ). b ) M< I(P) = (A ^ Ag, Bx, B3, D, D, Egg}. K P ro o f: a) D^ c o n ta in s no t r i a n g le s and no K uratow ski co v erin g y o f Dn has an is o la te d v e rte x in th e in te r s e c tio n o f th e two K uray to w sk i su b g rap h s. These f a c t s and Theorem 2.1 im ply t h a t D^ i s n o t th e so u rce f o r any S D -o p e ra tio n o f Types One, Two, I o r IV. I t i s m aximal and hence an i s o l a te d g rap h. F ig u re hj o f Appendix C shows I ( P ) \ fdq} i s connected and hence a component o f ( l ( P ), < v ). y n b ) A^, Ag, B^, By and D^ a re m aximal in ( l ( P ), < ). A case b y case check o f a l l S D -o p e ra tio n s w hich produce D ^ and Egg shows t h a t th e y a re maximal in ( l ( P ), < ). T his p o s e t ( l ( P ), < t r ) i s en co u rag in g f o r g e n e ra l p ro o fs because K

52 Uo DQ, found by N. R o b ertso n, h a s v e ry s p e c ia l p r o p e r tie s. I t i s m axiy mal in ( l ( P ), < ), i t c o n ta in s no subgraphs homeomorphic t o K^, and i t c o n ta in s d i s j o i n t k -g ra p h s. I t i s c o n je c tu re d t h a t th e p ro j e c t iv e p la n e i s th e o n ly s u rfa c e w ith a graph t h a t s a t i s f i e s th e gener a l i z a t i o n o f th e s e p r o p e r t ie s. T h is s u g g e sts t h a t analogous s p l i t t in g ty p e s f o r o th e r s u rfa c e s could g iv e connected p o s e ts. Since g + 1 2g + 1 U K and U K, th e u n io n o f K 's w ith a common 3 -c y c le, *3 5 5 are known to be ir r e d u c ib le f o r S and S r e s p e c tiv e ly, th e con- 8 g n e c te d n e ss o f t h i s p o s e t im p lie s th e K uratow ski C overing C o n je c tu re. The edges to be d e le te d a re p re s c rib e d in th e s e ty p e s so i t i s p o s s ib le th a t a com puter program u s in g th e se ty p e s would f in d th e s e t I(M ), S f o r sm a ll g, in a re a s o n a b le le n g th o f tim e. The program could ta k e two fo rm s. I f some c h a r a c te r iz a tio n o f l(m ) could be found th e n - K s t a r t i n g w ith th e s e graphs o n ly S D -o p e ra tio n s would be g e n e ra te d. These g raphs would be o rd ere d by subgraph r e la tio n s h ip s and em bedabil i t y would be checked from th e bottom, s m a lle s t g rap h, up u n t i l a nonp ro j e c t i v e graph i s found. A ll su p erg rap h s a re th e n known to be re d u - n c ib le. The o th e r p o s s ib le p la n would be t o s t a r t w ith U and u se S D -o p e ra tio n s and t h e i r in v e r s e s. K3 5 S ince th e in v e rs e s a re n o t w e lld e fin e d t h i s m ight be l e s s f e a s i b l e. However we make th e s e c o n je c tu r e s. C o n je c tu re A; Given a su rfa c e M, th e r e e x i s t s an a lg o rith m f o r g e n e ra tin g l(m ) which u s e s th e K uratow ski c o v e rin g o f th e elem ents o f I(M ).

53 C o n je c tu re B: com puter g e n e ra te d. The irre d u c ib le graphs f o r th e to r u s can be Ul We w i l l end t h i s s e c tio n by a g a in re v e r s in g o u r p o in t o f view and s tu d y in g a graph G, a K uratow ski c o v e rin g G = K U H w hich has v as an is o la te d v e rte x in K H H, and th e S D -o p e ra tio n r e s u l tin g in Sv G \ ( ( v ', v " ) }. P ro p o s itio n 2.6 : Given G 6 l ( P ), a K uratow ski co v e rin g G = HUK, v an i s o l a te d v e rte x in K D H, an S -o p e ra tio n whose e f f e c t on H and K i s Case ( l, l ), and d i s j o i n t k -g ra p h s A, B which a re c o n ta in e d in H and K r e s p e c tiv e ly, th e n S^K U S^H = sv g\ { ( v, v " )} i s n o n - p ro je c tiv e. P ro o f; The K uratow ski subgraphs H and K a re unchanged in Sv G \ { ( v ', v " ) } and hence s t i l l c o n ta in d i s j o i n t k -g ra p h s. Any graph which c o n ta in s d i s j o i n t k -g ra p h s i s n o n -p ro je c tiv e (se e P ro p o s itio n 2.5 [GH2] ). P r o p o s itio n 2.1'. Given G l ( p ), a K uratow ski c o v e rin g G = H U K, v an i s o l a te d v e rte x in K fl H, an S -o p e ra tio n whose e f f e c t on H and K i s Case ( l, l ), th e n S H U S K = S g \ { (v 1, v ")} can be i ) r e d u c ib le, i i ) ir r e d u c ib le, o r i i i ) p ro j e c t i v e. P ro o f; i ) U sing C^, th e graph

54 c o n ta in s F^ = ^ ^ < \ \ ( ( 9, 0 ), (3, 6), ( l, 5)}, (se e F ig u re 26 h2 and Appendix A page 9 7 ). F ig u re 26 S9 ( l, 2, 5 ) l. N {t9' 0 )) i i ) Any Type I s p l i t t i n g in A ppendix A shows t h a t S^K U S^H can be ir r e d u c ib le. 5,4 9 i i i ) U sing E_o, K and H j38 " y / ---- \ J '1 0 ; th e graph 8) K ^ Sl l ( l, 8) H = Sl l ( l, 8 ) E3 8 ^ 11 ^ embeds as seen in F ig u re 27. O I F ig u re 27 "38

55 S e c tio n 2.3 ( l ( P ), < ) and IN (p ) I n A ppendix A a l i s t o f a l l S D -o p e ra tio n s i s g iv e n. The Kurato w sk i c o v e rin g s used change from s p l i t t i n g to s p l i t t i n g s in c e th e req u irem e n t i s t o f in d a l l th e s p l i t t i n g s f o r w hich th e r e e x i s t s a K uratow ski c o v e rin g w hich w i l l make th e s p l i t t i n g o f Types One, Two, I, o r IV. I t i s n a tu r a l t o ask w hat i s p o s s ib le i f a s in g le Kuratow ski c o v e rin g i s chosen f o r each irre d u c ib le grap h in l ( p ). Many irre d u c ib le graphs have 10 o r more K uratow ski c o v e rin g s and i t would be no problem to p ic k K uratow ski c o v e rin g s w ith few good s p l i t t i n g s. However i t i s a ls o p o s s ib le t o p ic k good c o v e rin g s. D e f in itio n : U sing Appendix A, a s sig n each ir r e d u c ib le g raph th e K uratow ski c o v e rin g in i t s l i s t t h a t g iv es th e m ost s p l i t t i n g s o f Type: One, Two, I, and IV, i. e. f o r D ^ use D ^ = K^ U Hg. I f th e re i s a t i e u se th e f i r s t K uratow ski co v e rin g in th e l i s t, i. e. in Ag use th e c o v e rin g f o r Ag. The s p l i t t i n g s which u se th e a ssig n e d K uratow ski c o v e rin g and a re o f Types One, Two, I, o r IV a re c a lle d c - s p l i t t i n g s. D e f in itio n : ( l ( P ), < c ) i s r e f le x iv e t r a n s i t i v e r e l a t i o n g e n e ra te d by a l l c - s p l i t t i n g s. P r o p o s itio n 2.8 : ( l ( P ), < c ) i s a p a r t i a l l y o rd ere d s e t. P ro o f: < c i s a s u b re la tio n o f < and so i s a n tisy m m e tric. < i s r e f le x iv e and t r a n s i t i v e by d e f i n i tio n. c U sing th e s e K uratow ski c o v e rin g s th e b e s t p o s s ib le case i s found, i. e. I ( P ) \{ D _ }, {Dq } a re th e com ponents. S ince a n o th e r c h o ic e o f y y

56 co v e rin g s would g iv e d i f f e r e n t maximal elem en ts even i f th e components rem ain th e same, i. e. u se * D^q in s te a d of D g, i t i s u n p ro d u ctiv e to lo o k a t m axim al e le m e n ts. Theorem 2.3 ; T here i s an assignm ent o f a u nique K uratow ski c o v erin g f o r each graph i n l ( P ) such t h a t ( l( P ), ) r e s t r i c t e d to s p l i t t i n g s which a re o f Type One, Two, I, o r IV w ith r e s p e c t to th e a ssig n e d c o v e rin g s h a s two com ponents, l ( P ) \ {DQ) and {Dn ). y y P ro o f; F ig u re in Appendix C i s a su b set o f th e c - s p l i t t i n g s and ex am ination o f t h i s f ig u r e shows t h a t ( l( P ), < c ) has th e re q u ire d p r o p e r ty. In A2 > th e K uratow ski c o v e rin g i s two K ^ 's one o f w hich becomes a K^ ^ > h u t in B^. -> th e K uratow ski c o v e rin g o f i s ag a in two K ^ 's. T his shows t h a t f o r c - s p l i t t i n g s th e re i s no r e l a tio n s h ip betw een th e K uratow ski c o v e rin g s. U sing B^ and i t s c a n o n i- 2 c a l c o v e rin g which a r i s e s from th e d e f i n i tio n B^ = U K^ i t is re a s o n a b le to c o n s id e r a p r e f e re d K uratow ski c o v e rin g, t h a t in h e r ite d from B^. F ig u re 28 shows B^, B^, and w ith t h e i r K uratow ski c o v e rin g s in h e r ite d from B.^. T here a re two such c o v e rin g s f o r, b u t n e ith e r can be used f o r th e S D -o p e ra tio n from t o which s p l i t s v and would e f f e c t th e K^ as in Case S3, b u t would n o t d e le te any edges from S^,.

57 F ig u re 28 B2 ^7

58 These two exam ples show t h a t w h ile K uratow ski co v e rin g s can be h6 found w hich g iv e th e e n t i r e s e t l ( P ) \ (D^J th e y have l i t t l e r e l a t i o n sh ip to each o th e r. In a sk in g q u e s tio n s about ir r e d u c ib le graphs th e answ ers a re u s u a lly e a s ie r in th e c a se s where th e graph c o n ta in s d i s j o i n t k -g ra p h s (a k -g ra p h d i s j o i n t from an example f o r th e n e x t low er genus su rfa c e f o r g e n e ra l s u r f a c e s ). So in same sense th e m ost i n te r e s t in g graphs a re th o se which do n o t have d i s j o i n t k -g ra p h s. T h is s e c tio n ends w ith an ex am ination o f our su b p o sets r e s t r i c t e d t o th e s e g ra p h s. D e f in itio n : I N(p) is th e s e t o f g rap h s in l( P ) which do n o t have d i s j o i n t k -g ra p h s. D e f in itio n : (IN(P), < ), (ln(p), < K ), (IN(P), < R ) and ( I lf(p ), < ) a re th e r e f le x iv e t r a n s i t i v e r e l a tio n s on IN(p) gener a te d by th e S D -o p e ra tio n s from IN(P) to IN(p) which s a t i s f y th e r e s t r i c t i o n s f o r th e g iv en r e l a t i o n on l(p). P ro p o s itio n (IN(P), < ), (IN(P), < R ), (IN(P), < R ) and (IN (p ), < c ) a re p a r t i a l l y o rd ered s e t s. P ro o f: Each i s a s u b r e la tio n o f < on l(p) and hence a n t i sym m etric. a re p o s e ts. Since th e y a re r e f le x iv e and t r a n s i t i v e by d e f i n i tio n th e y Theorem 2.b: (ln (p), < K) i s a connected p o s e t w ith maximal elem ents A a n d Eg2. P ro o f: F ig u re k6 in Appendix C i s th e s e t o f s p l i t t i n g s which

59 g e n e ra te (ln (p), <^) and shows t h a t t h i s p o s e t i s connected w ith 3 maximal e le m e n ts. Since th e Type I and IV s p l i t t i n g s a re marked in t h i s diagram, i t a ls o p ro v es th e fo llo w in g theorem. Theorem 2.5 : (ln (p), <v ) h as two com ponents, {D_) and "" ^ ^ IN(P)\{D2). M< IN(P) = (A2, E22, D2, E2 ). K1 As s ta te d b e fo r e, p la y s a c e n tr a l r o le in l(p). P ro p o s i tio n 2.1 0, which i s proven by checking F ig u re U6, shows t h a t a v e ry sm a ll number o f s p l i t t i n g s, s ix, w i l l g e n e ra te any graph from B^. P ro p o s itio n ; a) For X IN(P) th e r e i s a sequence B, = X, X,,... X = X where X. and X., a re th e so u rce and 1 0* 1* n x l + l r e s u l t o f an S D -o p e ra tio n i n IN(p) o f Type One, Two, I o r IV and n < 6. b ) For X IN(P)\{D2 ) th e r e i s Y IN(P)\{D2 ) such t h a t y < K X and y < ^ B T his com pletes th e stu d y o f th e p o s e ts g e n e ra te d by "good" K urato w sk i c o v e rin g s. S e c tio n 2.k d is c u s s e s some r e s u l t s about th e i n t e r s e c tio n o f th e two K uratow ski subgraphs. S e c tio n 2.b I n te r s e c tio n P r o p e rtie s The p a r t i a l l y o rd e re d s e t (l(p ), < K ) n o t o n ly p re s e rv e s in f o r m ation about what happens t o th e K uratow ski c o v e rin g o f each so u rce g rap h, i t a ls o p r e c is e ly d e fin e s w hat happens to th e i n te r s e c tio n o f

60 th e two K uratow ski g rap h s which cover th e o r ig i n a l g ra p h. In th e case U8 o f Type One s p l i t t i n g s th e r e i s a d if f e r e n c e betw een w hat i s p o s s ib le and w hat i s observ ed in (l(p), ) F o r t h i s reaso n Type One s p l i t tin g s a re d is c u s s e d l a s t. Each component o f K Cl H f o r any K uratow ski c o v e rin g used f o r th e s p l i t t i n g s in (l(p), < ) has th e homotopy ty p e o f one o f th e g raphs in F ig u re 29- O l b. F ig u re 29 Homotopy ty p e s o f components o f K H H. T his can be seem from o b se rv in g th e i n t e r s e c tio n s in Appendix A. Any component o f th e i n te r s e c tio n which i s a t r e e i s.homotopic to 1, w hile th e o th e r th r e e graphs a re s u f f i c i e n t to c l a s s i f y th e r e s t o f th e i n t e r s e c tio n which a r i s e from th e K uratow ski c o v e rin g s used in (l(p), < K ) - In Type Two s p l i t t i n g s th e homotopy ty p e o f th e i n te r s e c tio n i s unchanged. T here a re th r e e c a s e s. Case l ) v i s n o t in th e i n t e r s e c tio n o f K and H. Hence e i s n o t in th e in te r s e c tio n o f S^K and s'"*!!, so KflH = S K fls H. Case 2) v i s in th e i n t e r - v 7 v v s e c tio n o f K and H, b u t o n ly one o f v ' o r v" i s i n th e i n t e r s e c tio n o f SVK and S^H. As in c a se 1, e i s n o t in S^K fl S^H so K fl H = S^K fl Sv H. Case 3) v i s in th e in te r s e c tio n o f K and H and v ' and v" a re b o th in th e i n te r s e c tio n o f S^K and S^H.

61 U9 Here we have e E(Sv K H Is^H). I f H S^H i s c o n tra c te d alo n g th e edge e th e n th e in te r s e c tio n i s i d e n t ic a l to K l~l H. S ince t h i s c o n tr a c tio n i s a homotopy e q u iv a le n c e, th e a s s e r tio n t h a t f o r Type Two s p l i t t i n g s th e homotopy ty p e o f th e i n te r s e c tio n i s unchanged i s shown in a l l c a s e s. By d e f i n i tio n Type I s p l i t t i n g s have a c o v erin g K, H w ith an is o la te d v e r te x, v, in K H H. T h is v e rte x i s s p l i t and th e new edge i s rem oved, le a v in g th e r e s t o f K 0 H unchanged. T h is shows t h a t Type I s p l i t t i n g s d e le te a component o f homotopy ty p e 1 from th e i n te r s e c tio n, K H H. In Type IV s p l i t t i n g s th e v e rte x t h a t i s s p l i t i s e x a c tly l ik e a Type s p l i t t i n g so any change in th e homotopy ty p e comes from th e new edge t h a t i s used to make K. As s ta te d b e fo re Type IV s p l i t t i n g s o c cu r when a v e rte x v i s s p l i t so t h a t v ' i s a tta c h e d to x^ and Xg> th en th e edge (x ^, x^) i s d e le te d. The c o v e rin g i s chosen so t h a t (x ^, x ^ ) i s n o t in th e in te r s e c tio n, hence i t s d e le tio n does n o t e f f e c t K fl stjh. The edge (v, x^) i s n o t in E(K fl H), b u t by d e f i n i t i o n (vf, Xg) i s in b o th E(S^H ) and E(k). Since v and xg a re b o th in V(K H h ), th e new edge, (v ', x^), adds a new p a th betw een v ' and x^. I f v and x a re in two components o f K fl H, th e n th e s e two components a re now combined in to th e same comp onent in K fl S^H. I f v and x a re in th e same component o f K fl H th e n i f i t h a s homotopy ty p e i in Kfl i t h as homotopy ty p e i + 1 in K fl Sv where i = 1, 2, o r 3. The f o u r exam ples a re o f

62 ty p e 1 to ty p e 2 and two components changing in to one component. 50 In Type I, Type I, and Type I, I s p l i t t i n g s th e o bserved changes in th e homotopy ty p e o f th e i n te r s e c tio n a re much more r e s t r i c t e d th a n th e t h e o r e t i c a l p o s s i b i l i t i e s. These changes a re l i s t e d in T ab les 1 and 2. In a Type I o r Type I, s p l i t t i n g e x a c tly one i s changed to a K and no edge o f th e i n te r s e c tio n i s d e le te d from th e graph 3 j 3 G. The edges which a re e lim in a te d from th e K,_, b u t n o t d e le te d from Sy G, change th e i n te r s e c tio n a g r e a t d e a l and i t can be seen t h a t th e s e edges acco u n t f o r a l l o f th e changes. L e t K be the w hich s p l i t s to a ^ an(^ (x^ X2^ 311 e<^6e which i s d e le te d from S^G. T here a re th re e c a s e s. Case l ) d (x ^) = d (x ^) = 4 in K. Case 2) d (x 1 ) = 4 and d(x2) = 2 in K. Case 3) = d(x2) = 2 in K. In case 1 th e i n te r s e c tio n rem ains unchanged as d (x ^ ) = d(x2) = 3 in S^K and so x ^ and x2 rem ain in S^K. In c a se s 2 and 3 th e r e i s an x^ ^ V(K H H) and x^ f. V(Sv K fl S^H), however t h i s im p lie s th e r e i s an edge (x ^, y ) w hich i s e lim in a te d from S K and n o t d e le te d from S G. The changes in th e in te r s e c tio n v v w i l l be e x p la in e d in term s o f th e s e edges. L et p be a maximal p a th o f edges and v e r tic e s (x, x ^ ), (x ^, x2 ),..., (xn, y ) w hich a re e lim in a te d from K b u t n o t d e le te d from S G. T his p a th i s a l l o r p a r t o f K_ in K and can be Tl ' V2 one edge o r a s in g le v e r te x. Since th e edges o f p a re n o t d e le te d

63 from Sv G we know p c E(K D H) and f o r each edge in p, 51 e f E(Sy K I"). I t i s c le a r t h a t x and y a re in V(S^H). There a re th re e c a se s as d e p ic te d in F ig u re 30, where s o lid l in e s r e p r e s e n t edges o f K and d o tte d l i n e s edges o f H. Case l ) x, y V(sJk). Case 2) x V( pk) and y V(S^K). Case 3) v, y f v fs^ K ). Case 1 Case 2 Case 3 F ig u re 30 P o s s i b i l i t i e s f o r p In case 1 th e r e i s one p a th betw een x and y w hich no lo n g e r e x i s t s. I f th e component o f K fl H t h a t x and y were in was a t r e e i t i s now two com ponents, o th e rw is e, i f i t was o f type i i t i s now o f ty p e i - 1, where i = 2, 3, ^. F o r case 2 th e r e i s a sequence o f edges (y, y1 ), (y^, y )... (y, z) where z 6 V(K) and t h i s sequence h as a t l e a s t one edge in i t. The edge (y, y^) i s d e le te d from G because i t i s n o t in th e in te r s e c tio n o f K and H, s in c e i f i t was p would n o t be o f maximal le n g th. I f y^ ^ z th e n y^ V(H) and th e r e s t o f th e p a th c o n ta in s a n o th e r p w hich i s h an d led s e p a r a te ly. The p a th p from x to y in K (1 H i s a sequence o f edges which can be c o n tra c te d to th e v e rte x x. T his does n o t change th e homotopy ty p e o f K fl H

64 and c o n tr a c tin g th e s e edges in K H H i s th e same as d e le tin g them 52 from S^K n S^H. So in case 2 th e homotopy type o f th e i n te r s e c tio n does n o t change. Case 3 im p lie s t h a t p i s an is o la te d a rc o r v e rte x in K D H w hich i s d e le te d from S K H S H. Hence S K H S H h as one l e s s v v v v component o f homotopy type 1 than K D H h ad. In summary th e homotopy type changes by lo s in g one c o n n e ctio n f o r each p a th in K th a t r e p r e s e n ts an e n tir e edge o f and i s e lim in a te d from s''"lt b u t n o t d e le te d from S G. The i n te r s e c tio n v v lo s e s one is o la te d component o f ty p e 1 f o r each p a th o f H which i s a c e n te r p a r t o f an edge o f th e and i s n o t d e le te d from G. F in a lly i f a c e n te r p ie c e o r h a lf an edge from K i s d e le te d from th e g rap h, SVG, th e homotopy ty p e o f th e i n te r s e c tio n rem ains unchanged, j u s t as in th e case where an edge o f K i s a c a n o n ic a l edge in th e Kc and i s d e le te d from S G. 5 v R ev ersin g th e p o in t o f view by ask in g w hat can happen when Y = n g iv e s th e fo llo w in g. I f Y = f, th e n th e r e a re two a rc s l i k e case 1 in F ig u re 21 and e i t h e r two components each change from i t o i - 1 homotopy type o r one component changes from i t o i - 2, where 0 and -1 mean two and th r e e components r e s p e c tiv e ly. I f y = 1, th en th e r e i s one a rc l i k e case 1 and e i t h e r a c a n o n ic a l edge o f i s d e le te d o r one a rc i s l ik e c a se 2 in F ig u re 21. T his means one component changes from i to i - 1. I f y = 2 th en th e r e a re two p o s s i b i l i t i e s. E ith e r th e re i s a case 1 and a case 3

65 w hich changes one component from i to i - 1 and d e le te s a ty p e 1 component o r th e r e a re two c a se 2's o r two c a n o n ic a l ed g es, h e re th e homotopy ty p e rem ains unchanged. I f y > 3 th e n a t l e a s t one edge o f K i s l i k e case 3 in F ig u re 21 and th e homotopy ty p e changes. T h is im p lie s t h a t o n ly y = 2 w i l l a llo w th e homotopy ty p e to rem ain unchanged. T able 1 g iv e s th e o bserved r e s u l t s f o r a Type I and Type I, s p l i t t i n g s. T able 1 Changes in homotopy ty p e f o r components o f f o r Type I and I, S p l ittin g s K fl H Number o f edges d e le te d Changes observed , , 2 + 1, 3 + 2, U 3 2 None + None, 1+1, 2+2, no i n te r s e c tio n For Type I, I s p l i t t i n g s n o t o n ly i s e v e ry th in g u nder Types I and I, a p p lic a b le, b u t th e edges e = (x, y) w hich a re in th e in te r s e c tio n o f K and H and a re d e le te d from S^G m ust a ls o be c o n s id e re d. F i r s t n o te t h a t th e s e a re i s o l a te d edges sin c e i f y i s o f v a le n c y two in b o th K and H a s w e ll as in K 11 H th e n i t i s o f v a le n c y two in G w hich i s im p o s s ib le. I f b o th x and y a re n o t o f v a le n c y fo u r in b o th K and H th e n th e re i s a p a th from x o r y to a v e r te x in H S^H w hich i s e lim in a te d from K o r H b u t n o t d e le te d from S G. The edge (x, y ) p lu s t h i s p a th o r th e s e

66 5^ p a th s i s an a rc o f K H H w hich i s d e le te d. T his means th e component which in c lu d e d x and y goes from i to i - 1, where i i s 1 to U and 0 means e i t h e r one component becomes two o r one compon e n t d is a p p e a rs. The p o s s i b i l i t i e s f o r changes in th e i n te r s e c tio n a re v a r ie d. F o r exam ple, i f y = 1 th en i f th a t edge was in th e i n te r s e c tio n one component goes from i to i - 1, w h ile th e r e a re th re e a rc s l ik e case 1 o f F ig u re 21. These th re e a rc s can cause th re e i to i - 1 changes o r one i t o i -1 change and one i to i - 2 change, etc. On th e o th e r hand i f th e edge in Y i s n o t in th e i n te r s e c tio n o f K and H, th e r e i s one a rc l i k e case 2 and th re e as in c a se 1, see F ig u re 30. The p o s s i b i l i t i e s a re so much more numerous th a n th e r e a l i t i e s th a t no l i s t o f p o s s i b i l i t i e s w i l l be g iv e n. T able 2 g iv e s the observed r e s u l t s. T able 2 Changes in homotopy type f o r components o f K fl H in Type I, I S p l ittin g s Number o f edges d e le te d Changes observed > 1, 3 -> 2 k 1 1, 2 -* 2

67 CHAPTER 3 (I(P), <) S e c tio n 3.1 M u ltip le s p l i t t i n g s betw een th e same graphs In tr y in g t o u n d e rsta n d th e p o s e t (l(p), < ) in term s o f th e K uratow ski c o v e rin g s i t i s im p o rta n t to f ir s t c o n s id e r th e fact th a t in many c a ses th e r e a re v e r t i c e s v and v f and s u b s e ts o f th e edge s e ts o f S G and S. G, Y and Y', such t h a t S g \ y» S. g \y '. v v' v v' In s e v e r a l c a se s t h i s homeomorphism a r i s e s from sym m etries in th e o r i g in a l grap h, see F ig u re 31. In th e r e a re te n v e r t i c e s any o f which could be u sed in th e s p l i t t i n g - d e l e t i n g o p e ra tio n to y ie ld C11. These te n v e r t i c e s a re sym m etric in th e sen se t h a t g iven two v e r t i c e s, x and y, th e r e i s an isom orphism from o n to A^ which in te rc h a n g e s x and y. T h is kind o f m u ltip le d e te rm in a tio n o f a s p l i t t i n g does n o t cause d i f f i c u l t i e s sin c e th e same isom orphism which in te rc h a n g e s th e v e r t i c e s w i l l g iv e an isom orphism betw een p o s s ib le K uratow ski c o v e rin g s. 55

68 56 F ig u re Symmetry i s n o t in v o lv ed in e v e ry m u ltip le d e te rm in a tio n o f an im m ediate r e l a t i o n in l(p). In F ig u re 32 a la b e lin g i s shown f o r U sing t h i s la b e lin g Fy i s reach ed by b o th S2j _(3 5 6 ) E13^ {(6, 10)} and S10^g X {(11, 9)} U sing th e la b e lin g s o f F ig u re 33 th e e q u a lity o f th e s p l i t t i n g s can be observed by u s in g a v e rte x mapping from S1Q^ ^ E ^ X {(9, l l ) } to S1;L( 3^ ^ g) E1 3 \ ( ( 6, 10)} as fo llo w s : 3 ^ , Since th e v a le n c y o f 11 i s 5 and th e v a le n c y o f 10 i s U i t i s c le a r t h a t no symmetry w i l l e x p la in t h i s m u ltip le d e te rm in a tio n o f E ^ Ey L ik ew ise, th e r e i s no correspondence betw een p o s s ib le K uratow ski c o v e rin g s o f E 13 '

69 57 F ig u re 32 E, F ig u re 33 {(9, 11)} In se ek in g a c l a s s i f i c a t i o n by K uratow ski c o v e rin g s o f each s p l i t t i n g in (l(p), < ), m u ltip le d e f i n i tio n s a re ig n o re d. F o r e ach H < G, th e " b e s t" v and Y a re u se d. Here " b e s t" i s d e fin e d by a s p l i t t i n g which i s o f Type One, Two, I, IV, V, orvi e x i s t s. Type V and VI a re d e fin e d in s e c tio n 3.2. P r e f e r ence i s g iv en to th e f ir s t f o u r, b u t o f th e lj-7^ s p l i t t i n g s o n ly 2^7 a re o f th e s e ty p e s. E xcept f o r 2 s p l i t t i n g s which a re d isc u s s e d in s e c tio n 3. 3, th e rem ain in g 225 s p l i t t i n g s a r e d iv id e d in to two m ain c la s s e s. Those whose K uratow ski c o v e rin g s keep th e same v e r t i c e s and th o se where th e K uratow ski c o v e rin g o f th e r e s u l t i n g

70 graph u se s b o th new v e r t i c e s as v a le n c y th r e e in a new K, which does n o t a r i s e from a K,-. 5 S e c tio n 3.2 Types V and VI D e f in itio n ; An S D -o p e ra tio n changing G t o S^gXy i s a Type V s p l i t t i n g i f i t i s n o t a Type One, Two, I, o r IV s p l i t t i n g and th e r e e x i s t s K uratow ski c o v e rin g s G = K U H and S^gX y = K U H such t h a t ; i ) st k = K. i i ) V(H) = V(S^H). Type I, V I, Type, V, etc. w i l l be used t o in d ic a te what i s th e e f f e c t o f th e S D -o p e ra tio n on K. T his ty p e o f s p l i t t i n g o ccu rs n e a r th e m aximal ele m en ts where th e r e a re m u ltip le p a th s betw een two v e r t i c e s. I t i s f r e q u e n tly th e new ed g e, e, which i s d e le te d. In F ig u re 3^ th e s p l i t t i n g B^q = g^a^x {(0, 11)} i s shown. The ex p e cte d c o v e rin g K As seen h e re th e s e s p l i t t i n g s can r e s u l t i n a change in th e homo- topy ty p e o f th e i n te r s e c tio n u s u a lly to h ig h e r c o n n e c tiv ity.

71 sl l ( l, 6 ) \ F ig u re 3if- B10 Sl l ( l, 6 ) \^ ^C11* Bg -> G i s an example o f two K^'s changing to two s w ith th e edge ( 9, 0) b e in g re p la c e d by th e p a th s ( 9, 8 ), (8, 2), (2, 0) and (9, 7 ), (7, 5 ), (5, 0). T h is g iv e s 1 = ( g / 9 / 3^ 7 \. / J / 5 0 l \ in s te a d o f y o '! 6 5 / 3X1(1 B = V / in s te a d o f os:;) as shown in F igure 35.

72 6o "N Br f r S9(5,6,8)B8X <3-8) (5,6 ), (It, 7 ) ) G= S9(5, 6, 8) * 8 ^ * ). (3 8>< (5, 6), (4, 7 ), (0, 9 )) F ig u re 35 In Bg > th e new edge i s n o t d e le te d, h u t th e edge t h a t i s d e le te d c o u ld be re p la c e d by a t l e a s t 30 d i f f e r e n t p a th s. As shown in F ig u re 36 th e two K s a re s tr e tc h e d and th e n th e edge 5 (2, 3) i s re p la c e d by th e p a th s (2, 8 ), (8, 7)* (7, 3) and (2, 6 ), (6, 5 ), (5, 3).

73 S2(l, 3, 6, 8)B2 F ig u re 36 The p re v io u s exam ples have v o r v ' as one e n d p o in t o f th e e x tr a d e le te d edge, b u t in ^ 5) B6B A {(2, 3 ), (3, U) (1+, 5 )) th e edges (3, and (2+, 5) a re d e le te d le a v in g th e p a th (1, 5 ), (5, 3) in s te a d o f th e p a th ( l, 5 ), (5, *0, 3). The o r i g in a l p a th i s n e c e s s a ry to in s u re t h a t K U H = Bg. F ig u re 37 shows t h a t in t h i s example th e homotopy ty p e o f th e i n te r s e c tio n rem ains th e same.

74 62 3, Sl ( 2, 5, U) b6^ ^ 2* 3 ^ F ig u re 37 E20 Sl(2, 5, U) (3,M, (*, 5)) 3^j Type VI s p l i t t i n g s, w h ile b e in g th e m ost num erous, a re th e l e a s t d e s ir a b le, sin c e one g f r ia th e o r i g i n a l graph d is a p p e a rs and th e m ost t h a t can be s a id o f th e Kg g w hich r e p la c e s i t in th e new graph i s t h a t b o th v 1 and v" a re v a le n c y th r e e v e r t i c e s. T h is ty p e can n o t in any way b e u se d t o d eterm in e th e s p l i t t i n g. A com bination o f th e b e s t v and Y w ith th e b e s t K uratow ski c o v e rin g to g iv e one o f th e s ix ty p e s i s u se d and th e c h o ic e s a re made so t h a t

75 Type VI i s used o n ly when i t i s f o rc e d. T h is o ccu rs m ost fre q u e n tly n e a r th e m inim al elem en ts when th e so u rce graph has o n ly a few p a th s betw een two v e r t i c e s and i s th e u n io n o f two K,, s. Jj J D e f in itio n : An S D -o p e ra tio n changing G to S ^ g Xy i s c a lle d Type VI i f i t i s n o t Types One, Two, I, o r IV and th e re e x is t K uratow ski c o v e rin g s G = K U H and g\ Y = K U H such t h a t : 1. K = /S~K v 2. IT i s homeomorphic to K and h as v 1 and v" as 3 ) -J v a le n c y th re e v e r t i c e s. Type I, V I, and Type, VI w i l l be u se d to in d ic a te w hat th e e f f e c t o f th e S D -o p e ra tio n i s on K. Two t y p ic a l exam ples o f Type VI s p l i t t i n g s a re shown in F ig u re s 38 and 39. In -* F^q > S^K i s i d e n t i c a l to K, w hile in Fq -> G, one edge o f K i s su b d iv id ed in S K. These two y * b a s ic k in d s o f Type VI s p l i t t i n g s make up o v e r a t h i r d o f a l l o f th e s p l i t t i n g s.

76 s i ( 2, 9, 10) E15 p10 = ^(2, 9, 10) 6)1 F ig u re 38

77 F, F ig u re 39 G 37<6, u ) F9 X t(5 10)) W hile m ost Type VI s p l i t t i n g s in v o lv e o n ly ^ ' s some use a K,- and a K0 _. F ig u re lj-0 shows -> Dnr7. H ere th e K-., 5 3, ^ ^, changes t o a 3, "Cg Q 3{ 2 ) > 111 ^he norm al way, how ever H = ^ 3 6^ o 4 ^2 h a s no r e l a tio n s h ip t o th e o r i g i n a l h-g b 5 >6) These subgraphs a re chosen p r e c i s e l y b ecau se

78 K U H = Ag and K U H =. Where p o s s ib le edges w hich a re e lim in a te d from a and n o t d e le te d from th e g rap h, i.e. (U, 5), a re in th e i n te r s e c tio n o f K and H, b u t edges o f Y t h a t do n o t a r i s e from a K,- changing t o a K- Q a re in t h i s i n t e r s e c t i o n. A, Sl ( 7, 2, 3) A2^ 2' F ig u re i+0 D17 Sl ( 7, 2, 3) A2^ (3, U ), (5, 7 ) ) 2' 6 ^j S e c tio n 3.3 E x c e p tio n s. Of th e hfk s p l i t t i n g s, a l l b u t two have v e r t i c e s, edge s e ts and K uratow ski c o v e rin g s w hich a llo w th e s p l i t t i n g to b e c l a s s i f i e d in to one o f th e p r e v io u s ly d e s c rib e d s ix c a te g o r ie s. The two

79 e x c e p tio n s a re -> Fg and ->. 67 F ig u re i+l CY > W ith th e, la b e lin g i n F ig u re h i, Fg = 2 g) ci,. \ f ( l, *0# (3, 6 ), (0, 9)}. I n th e r e s u l t i n g grap h, Fg, i t i s im p o ssib le t o f in d a K-, - w hich u s e s b o th 0 and 9 as v a le n c y th r e e v e r t i c e s..j ;.j I t i s a ls o im p o ssib le t o cover w ith two g '3 where one o f them does n o t u se ( l, h ) o r (3, 6 ) o r would need to u se (0, 9) These two f a c t s can be v e r i f i e d b y a case by case a n a ly s is. The b e s t d e s c r ip tio n, u s in g th e K uratow ski c o v e rin g s K = ( 4 < i I 6) H and i)> 5 = C t VLs) ' i s t h a t b o th K3 3' s move and e ach new K3 3 u s e s one o f th e new v e r t i c e s as v a le n c y 3. I n a se n se th e se two v e r t i c e s a re c o m p letely s e p a ra te d from each o th e r. The o th e r e x c e p tio n, F1 = ^ ^ is th e

80 o p p o s ite problem. I n s te a d o f r e s t r i c t i o n s on th e K uratow ski c o v e rin g o f th e r e s u l tin g g rap h, th e r e i s a unique c o v e rin g f o r, (8 9 7 \ (2 b 6 \ K =l1 g 10/ > H \ 1 3 5/ * T h is spl i t t i n 6> w hich i s th e o n ly p o s s ib le way to g e t to d e s tro y s b o th o f th e e x p e cte d S^K and S^H. U sing K = ^ i and ** = ( 7 ^1 2 6 ) a d e s c r ip tio n would be t h a t b o th new K_ 's u se th e two new v e r t i c e s as c a n o n ic a l v e r t i c e s o r t h a t th e s e two need to be u se d to g e th e r. JO F ig u re k2 D. S e c tio n 3.^ C onclusions In c o n s id e rin g th e n a tu r a l p a r t i a l l y o rd ere d s e t (l(p), < ) and th e K uratow ski c o v e rin g s f o r each ir r e d u c ib le grap h, i t seems c l e a r t h a t th e s e two id e a s a re r e l a te d. W hile many s p l i t t i n g s in (l(p), < ) can n o t be p h rase d in term s o f a K uratow ski c o v e rin g enough can be to j u s t i f y t h i s s ta te m e n t. The v a rio u s su b p o sets o f (l(p), < ) have s e v e r a l u se s and Appendix A can be used t o t e s t g e n e ra l h y p o th eses f o r th e s p e c if ic case o f th e r e a l p r o je c tiv e p la n e.

81 APPENDIX A T h is t a b le g iv e s th e K uratow ski c o v e rin g f o r each s p l i t t i n g in th e g e n e ra tin g s e t o f (l(p), < ). Each ir r e d u c ib le g raph in l(p) i s l i s t e d w ith a diagram and la b e lin g. F o llo w in g th e diagram, each s p l i t t i n g i s g iv en w ith th e K uratow ski c o v e rin g s o f G and g\ y. In a d d itio n th e ty p e o f th e s p l i t t i n g and r e p r e s e n ta tio n s o f th e two in te r s e c tio n s, K H H and K fl H, a re g iv e n. F o r ty p e s I, n, rsj and I, K and H a re s e lf - e x p la n a to r y g iv e n th e s p l i t t i n g and hence a re n o t e x p l i c i t l y w r itte n down. I f a K uratow ski graph i s u sed more th a n once as a subgraph o f a given ir r e d u c ib le g rap h, i t i s la b e le d th e f ir s t tim e and r e f e r r e d to by i t s l a b e l from th e n on. F or convenience th e same l a b e l s a re u sed from irre d u c ib le g rap h to irre d u c ib le graph b u t th ey have no r e l a tio n s h ip to each o th e r, i.e. K^ u n d er A^ i s d i s t i n c t from K.^ under A^. 0 u nder th e type means t h a t th e r e i s no change in t h a t K uratow ski g rap h. Most o f th e s p l i t t i n g s i n th e s e ta b le s w ere f ir s t found by C.S. Wang [Wa] in h is w orking p a p e rs. TABLE 3 KURATOWSKI COVERINGS FOR (l(p), < ) 69

82 70 New ^ ^ Type K D H / Graph S p l i t K /H K /H K /H k H h A3 9(2, 3,b ) 1 2 K, = 3 b ^ = A5 9 (1, 2, 3, *0 K± I * \((9, 0)} H1 B8 9 (1, 3, b, 6, 7, 8) Kx l 2c V \((9, 0)J 3 ^ 9 6 H1 8 V. O ^ V B10 9(1, 3, b, 8) K± l 2o7 \((9, 0)} 3^9 H1 5 ^ ^ 9 (1, 2, 3, b, 6, 8 ) Kx 0 V(5,7), (6, 8)} Hl 1 D13 9(1, b, 5, 6, 7) ^ U 0 8 \ { ( 1, b)f (2, 3), (0, 9)} 2 3 9V v =L 5 %* V 7 6 8

83 New Graph S p l i t K / H K / H Type K / H 71 Kl"lH/ KflH f 6 9 (2, k, 5, 7) \ { ( 1, 3 ), ( 2, 4 ), ( 5, 7 ), (6, 8), (9, 0)} K, H 1 3 9, i B? 2( 1, 3, 7) \{(1+, 6)} 1 2, 1+ 6 v^5 7 >3 1 5 ' >6 3 l+'-w' 2 II c? 1(1+, 5, 6) M (2, l+), (5, 7)) 7 ^ 2 I n- 6 5 II , VI D3 1 ( 3, 1+, 5, 6) 5 l \ { ( 2, l+), (2, 7 ), (5, 7)} 1+ 6*^3 II 7 6 U, > VI

84 72 New Graph S p l it k / h Type kdh/ k / h k / h frnir D 1 (2, 3, 7) \((2, 6), (3, *0, (5, 7)} k \ 7 2 ' - * ' : / 7 2 VI // E3 1 (2, 3, 5, 6) \{(2, 3), (2,6), (3, 5), (5,6)} 1 7. >3 2T O r5 ^ 7 ^ 3 Tl U o I VI V* El8 1(2, 4, 7) 7 2^3 Y 4 \ { ( 2, U), (2, 7), (3, 5), (5,6)} 1 7, 1 >2 10(6, 8, 9) 2 \ K± = U Hx = 9 8 7

85 New _ ^ Type KHh / Graph Split K /H K /H K /H SfHff 73 A5 10(6, 7, 8, 9) ^ III V(0, 10)} H1 B10 10(5, 6, 8, 9) Kx 2 1^79 V \ { ( 0, 10)} 3 I). ^10 8 7, V Q *5 C6 2(1, 3) Kx I V (l, 3), (*, 5)} H 0 Cg 10(1, 5, 6, 9) Kx 0 \{ (6, 9), (7, 8)} Hl 1 C1]L 2(1, 5) Kx \{(1, 10), (5, 10), (3, U)} Dl8 10(1,7,8) Kx 2 1 V \C(6,9), (7, 8), (0, 10)} 3 b 5 V j V O

86 7^ New Graph Split k / h Type k H h / k / h k / h S nff a5 11(1, k) \((o, 11)} 2 1 > 11 ^ III 7 6 Hii = B10 11(1,6) \((0, 11)} K, 2 3*6a 5 3 ^ ,1 \ V 0 C10 8(6, 9) \(6, 9), (7, 10)} K, C1;L 8(6, 10) \{(6, 11), (11, 10), (7, 9 )} K

87 New Graph Split K /H K/H Type K/H k Dh / n r 75 cu 10(6, 9) V(6, 9), (7, 8)} ^ B, B2 7(1, 2, k, 6) 2 7 K1 = l 3 ^ 7 6 ^ = ^ 1 5 II II 0 6 Bu 7(1, K 6 ) 2 7 k2 = 3 l v II o H II o b5 7(1, 2, 6 ) 2 7 >5 II o H2 = 1 5 V 3 II o

88 New ^ ^ Type K H H / Graph Split k / h k /H K /H K H h 76 7(1, 2, 3) K,. 7 1,6 2 3 r II H V' D3 7(1, 2, 3, 6 ) K, II o \ { ( l, 6 ), (b, 5)) o E3 7(1, *0 \f(l, b), (5, 6 ), (2, 3 )) K. o B6 3(1, 2, 7) II 6 K1 = ,4 11 H = df C3 3(2, 5, 7) & \f(2, 3)) Kp = ^ 3 2«** i 5 i\fi H2 = 7 3 ^ x 6

89 New, ~ TyPe KOh/ Graph Split K /H K /H K /H STDS' 77 2 (1, 3, 6, 8 ) 5 3. ^ 5-3-2, ^ 2, \{(2, 3)) K 3 = ?-3 6^-3, 8 ^ 3 ^ 5 V $ 1 >6 1 7-^2 o D3 2(3, 5, 6, 8 ) k3 II \ ( ( 1, 7), (*, 8)) 6 D, >7 2(1, 3, 5, 6 ) l i - 3 ^ 0 & \ ( ( l, 3, (7, 8 )) k, = H1 6 Dg 2(3, 5,6, 7) K3 i i 6 - V(3, 7), (1,8)) H1 6 Eu 2(5, 6, 7, 8 ) K3 i \ ( ( l, 3 ), (5,6 ), (7, 8 )) H1 E (1, 6, 7 ) V (3, 5 ), (1, 6 ), (l, 7 )) Kc.H V V- 1 5 H3 = 2 3 7^ I &

90 New Type k Hh / Graph Split K /H K /H K /H K 0 H E22 2(1, 5, 7) K V 6 V(0, 3), (1, 1), (1, 5) H1 6Y ' 8 2 V P1 2(5, 7)V(1, 3), (7, 8), (5, 6), (l, 0)} b 2 1 Kg = // Hn B B8 1(2, 3, h) 7 1 II o K1 = 'O C2 3(1, 5 )\((2, 4)} K± 0 O 1(2, 3)\{(2, 3), (U, 5)} Kx II o I o

91 New Graph Split K /H Type K fl H / K /H K /H KflH D5. l( 4,6 ) \{ ( 2,3 ), (4,5)} K, II i o o F1 1(4, 6, 7)\((2, 3), K, I o ( ^ 5), (6, 7), (5, 8)} H I 0 B6 4(6, 8 ) 2 Kx = 5 6 l o 3<*-8 Hi = 7 2n x II / B9 1(2, 3, 6) K, M i II ft 3-^8 H2 = lt lft 2 II D6. 1(7,8)\((2, 3), (7, 8)} 5 6 K- = l-r-4-^ II o

92 New ^ ^ Type K H H / Graph Split K/H K/H K/H k Dh D? 1(5, 8)\{(2, 6), (Ifr, 5)) Kg I O H3 I! o Dq 1(2, 5)\{(2, 5), (b, 6 )} I 0 O h3 II ^ D12 1(2, 3, 7) 5 1>,7 I \((2, 8), (5,6)} Kr 6 2 ^ + >3 8 ^ 1 2. II E, 1(2, 8)\((2, 8), (3, 7), K I O (5,6)) V'* H3 e5 1(2, 7)\{(2, 7), (^, 8), I O (5,6)) / H3 e19 1(3, 7, 8)\f(l, 7), k 3 ii T (2, 8 ), (2,3 )) H5 = 2 H l ' e21 1(3, 5, 8)\{(2, 6) 5 2 I O (2, 7), (^, 5)) K6 = 6 l-jt.

93 81 New Graph Split K/H T>?e s s / k / h k / h k n h 1(2, 3, 8)\{(l, 2), (2, 7), (2, 8), (5, 6)}. i l 3 8 VI Bc B9 2(5, 6, 8) 1 6): K1 = 5 2T ^ i i o % II Hx = 7 l T 3 C3 U(l, 3)\{(2, 3)} 1 6*3 k2 = 2 5V 8 n i i 3 5 E2 = b o" l VI C. 2(1, 3,6, 7 )\{ (2, 3)} 1 % K3 = 5 6T U H3 = 2 * VI <J

94 Split k / h / v f /S» K /H Type K / H KflH/ k Hh 2(1, 3, 7, 8) \{ ( l, 10, (5,6)} K, I 0 o O 2(5, 6, 7, 8) \{ ( i, U), (5,6 ), (7, 8)} 6 l o 056 v Kk = 2 5w V, ^ 2 1 VI H^ = * & 2(6, 7, 8)\((l, 5), (1, 8), (7, 8)} II * 8 W H5 = 1 7 b 7 >2 5 VI 2(3, 5, 6) \f ( l, 5), (1, 10, (2, 6)} 2 W * k6 = ,c V > 5 7}^ V 8 H/- = 6 7 Ij- 1 2!< VI es' 1(2, 6, 7) \( ( 2, 6), (2, 7), (0, 4)} I

95 New Graph Split K/H Ty?8 K /H K /H k Hh / k n h 83 F-L 2(3, 5, 7)V(1, 5), (1, 8 ), (0, 6 ), (3, 7)) 2- H 6f *3 K = l 2 Ik 6 VI 6* 4 avi = (J B, d? 1(3, u, 6, 7) K = 3 5-»*2 > II * cr H1 = Dq 1(2, k, 6, 7)\{(3, 5) K, I & (S 9)) II <$ Du 3(1, t, 5, 7) \{(1, 7), (2, 6 )} K2 = n 11

96 New ^ ^ Type K A H / Graph Split K/H K /H K /H {cdh 8h D15 1(5, 6 )\f(0, 2), (3, 6)} 2 ^ 7 V 2^9 V ^ K3 = 6 1V >28 11 H_ = 3 5 J 9 & E5 6(2, 7)\{(1, 3), 5 2 lk 0 5), (7, 9)} KU= y 6^5,4 I ' 9 = E? 2(1, 3 )\((1, 3), 5), (6, 7)) K5 = 1 2'+-54 // A t 9 = II E20 1(2, 5, M V (2, 3), K V 6 (3, M, (U,5)J ?V2 ^5 h6 = S21 1(2, U, 7)\{(3, 5), (3,6), (U, 9)} 6 K, = VI

97 New Type khh/ Graph Split K /H K/H K/H k Hh 85 e23 1(2, 6, 7)V(l, 6), 9>^ 1 II Jt (3, 7), (3, 2)} K = P \? m Hg = U 7 «V 5 VI E28 1(2, U, 6)\((2, 6), (3, 7), K6 II (3, 5)) 5 \ a ^ 0 1 ^ 8 * VI F1 1(3, k, 5, 6)\{(1, 3), (2, 8), (3, 6), (If, 5)} K V O H. if 5 o F3 1(2, 3, 7)\{(1, 3), (2,3 ), (3, 6), (if, 5)) K, H F5 1(2, 3, if, 7 )\((2, 3), (3, 6), (if, 5), (1,3)} K, 1 v 7 0 2*8 / / l J v 5 6 VI & 7 2 0

98 B 86 New _ ^ Type KfiH/ Graph Split k / h K /H K /H KflH C3 2(1, 3)\((1, 3)) 1 2 ^ I 6 * 7 3 ^ 5 5 3*2 0 * >1 7 H 6 5(1, 3)\ {(1, 3)) l 6^ 7T 5 32 y t. 7 n / / D? 1(2, 3, 5)\{(1, 3), (7, 6 )) 1 6J V > ; ^ 7 V & i i E20 7(2, b, 6)\{(l, 2, ) i - * - 7 II +1 (3,^), (5, 6)) 3 2 ^ ~ VI 3s >! '4 h

99 New _ ^ Type KflH/ Graph Split K/H K /H K /H Kflff 87 E21 1(2, 5 )\{ (3, 7), 7 1 J I YL < >, 5), (6, 7) r 1* 1 7 5, VI / (2, U, 6)\f(l, 3), (3, «0, (1, 6), (5, 0)} 2,6 4 8 II 8V l VI O 6f Be B10 9(1, 2, 8) 5 \3_ II O Kx = l 3 4 O 37 = C5 8(3, 9)\{(1, 2)) Kx 0 0 ^ I Y ' 8(1, 9 )\{ (2, 3), (l, 9)} Kx II O I O

100 New Graph Split K /H K/H Type K/H KHh/ k n h \ 9 9(1, 5, 6, 8) \ { ( 1, 8), (2, 3), (0, 9)3 K1 s 5 1*f T 8 6 7$f9 V O H1 W V G 9(5, 6, 8)\{(1, 2), (3, 8), 5 \ a (5, 6), (1*, 1), (9, 0)} u, 1 V i _ \ ro^o CO H V 0 B1X 9(1, 3,4) 1 -^ 3 II = 8 9-J-2 0 9j ^ 7 Hi. k 5T ( 3, M \ ( ( 3, ^ ), (5,6 )} Kx 0 O

101 New ^ ^ Type k Hh / Graph Split K/H K /H K /H KHh 89 D15 9(1, 5)V(5, 6), 2 V -\a9 11 ^ (1, 2)} 8 ^ 3 ^ 5 1? a 5 5? ^ E10 7(9,3>\t(3,9), 7 II (1,8 ), (It, 5)) «: J U ) W A " 4 ' ' " E12 9(^, 8) \ { ( 2, 8),,2 6 9 <. (3, 7), (^, 5)) , h VI ^ 6 C V ^5 ) 2 *6 7 ' e21 9 (1,3,8 )\{(1,8), y ; II V,. (E, 5), (5, 6)) K2 = 6 g 3 w 9V\> ^ H2 = E h 9(1, 5, 7)\((1, 2), 6 A 8 i i A. (3,0 ), ( ^ 5)) 2 A 1 8^ 7 ^ 5 8? 2^ s A l A - V VI

102 New ^ Type KHh/ Graph Split K /H K /H K /H k Hh E27 9(3, 7)\((1, 8), K2 II (3, 9), (*, 5)) h2,$. A «9 vi!< K 9(^, 8) \{ ( l, 2), (0, 9), 9 6 8, 0 -V (3, 7), (*+, 6)} 7 l it 0 1 t 'S? > 6 7 >1 8 X it VI 0 F3 9(S 7, 8)\{(l, 8), " M s 5 (2, 3), (^, 5), (7, 9)) 6 ^ i VI 2< W 5 7 8* 51 8 ^ it 7 7 o it- F5 9(^, 7, 8)\{(l, 8), 6i 3 it 8 II ^ (2, 3), 6), (7, 9)3 7-^2 9

103 New Graph Split K / H ( V. / v K / H Type K H H /, (V - ( V ' K / H K H h c9 10(9, 6) \ f (7, 8)) > K1 = 3 ^ 5 8 9i3 Hx = ^5 0 0 Dl8 10(9, 8)\ ((9, 8), (6, 7)} K, II 0 I o 'i i D M3, 10)\{(3, 10), 10 1 ^ 9. 9 (5,6)) 8 % 0 O b 6 s-* o F3 M3, 5)\f(l, 10), 1JL3 (2, 5), (2, 8), (7, 9)} k, = 10 II

104 92 New Graph Split K/H rs/. f v K/H Type K/H k Hh / k n h f9 1(2, 10)\{(lf, 6), (7, 8), (5, 9), (3, 10)} Ki i 9 > v VI ic-f1? / 5.o < y } s c :6 5(1, 2, 3, 8) 2 1 Kx = 3 ^ 5 II CQ 5(2, 6, 8, 10) K. II 0 cu 5 (1, 2, 3, ^M (0, 5 )} K, III H D13 5(1, 3, h, 6, 10) \{(0,5 )} K, o

105 New Graph Split K /H ~ ^ k H h / K/H K/H KflH 93 di8 5(1, 3, 10) \ [(0, 5)} K, 1 v 3 4 5* V o 3(1, 2) \ ((1,2 ), (4, 5)) K, I 0 5(8, 10, l, 3 )\C (1, 3), (2, U), (0, 5)} K, lo / 2 7Sb 1 V 7 9 % * V O c5 1(7, 8, 9) >< II / ^ = 6 ^ 2 D, 7(1, 3 )\{ (8, 9)} K, I 0

106 9^ New Graph Split K /H rw K/H /-v» Type K/H k Hh / k Hh 1(2, 7, 8, 9)V(2, 3)} K, II VI 0 l b 6 D_ l(u,6,7,8)\{(2,3)} K, f0 U 6 II vi ^ G e6 1(2, k, 6, 9) \((7, 8), (3, 9)) I 0 Eg 1(2, k, 7, 9) \((7, 9), (8, 3)} K, I II F1 i(k, 6, 9)\((2, 3), (3, 9), (7, 8)} t? I VI

107 95 New Type K fl H / Graph Split K/H K/H K /H fc H fif D2 ) f J / 9 IV <{ 2 5 ^ 9^ ^ X 7 { \ 9 \ Qr Din 10 6(3, 5)V(3, 5)5 5 ^ -l2 I ^ Kx = 3 b S Hl = 3 7 V II _ * E 3(1, 2, 5, 6)\{(1, 2), 2Byi?v56 0 (5,6)} K2 = T n 9 3a VI 4* y+6 8t H0 = 8 r ^ E23 3(1, U, 5, 6 ) \{ ( l, 3), K2 2J 39^87 VI ^ U.M ) 5o 5^ 7 H2 II < E25 1^(2, 3, 6 )\{(2, 3), (3, 6)} 8 3 6s 2 v, v 3 y * lr l 5

108 New Graph Split K/H / v. <v K /H Type K/H k ^ h / KHh E2q 3(1, k, 8)\{(1, If), (5, 6)} K, I ki- 9 M 3 V T II k 3(2, 6, 8)\C(6, 14-), (1,*), (0, 5)3 2 y? r < 6 a7^ h ^ 3 VI % ^. V II & d12 9(1, 2)\{(U, 7)} K1 = 9 A 3. 2-y-i V!?>< 6 9 II / d15 9 (i,^ )\f(l,m ) I 1C H 0 * E25 l(^, 5 ) \ ( ( 2, 9), (3, 9)} K, * k.k *5 # k 5 <5'- 9 1 ^ * 7 67 l 9

109 New ^ ^ Type KHh/ Graph Split K/H K/H K/H k Hh 97 F 9(1, 6)\((2,»f), ^ (3, 5), (0,9)) 9 " 6 8 t i i W F 9(1, 2, 5)\((0, 9), 4,8 9 4,8 9 V (3, 6), (1,5)) K2 =7K 67K 4,96, 4 Jo 5 VI mss V n H2 = 2 3 V 'o'z7 9 d F6 9(1, 2, 3)\{(1, 4), (9, 0), k * (6,3)} H2 0 M G 9(1, 2,4)\{(0, 3), 9 3 II ^ (9, 4), (l, 5), (2, 6)} k < 9 b i # k 9 71 e *Exception

110 New Type K H H / Graph Split K /H K/H K /H k Hh 98 Cg 10(7, 8, 9) 7 6^3 / ^ = ^ 10 0 / ^ = D13 10(5, 7, 8, 9) 7 6^3 II \f(3, 5)) K2 = H1 o it 3,6 vi C5 JB 5 l 10 \ 8(7, 9 )\((3, 6), (7, 9)) ^ I H 0 E13 10(1, 2, 5, 7) Kx I \{ (6, 7), (8, 9)) Hl e (1,2,8,9 ) k2 6 - ^ V ^ \ f (3, l), (0, 10)) ' Hl 3 10 k VI & 7/ F2 10(1, 2, 8, 9) ^ 8 9yj5: V(6, 7), (8, 9), (0, 10)} d 3 ^ 109 VI

111 99 New Graph Split K/H / v, / v K / H Type K /H KDh / / v _ <v KHh G 10(2, 5, 7 ) \{(0/10), (8, 9), (1,3), (6,7)) K, 9 V U VI C10 11(1, 6, 8 ) 8 7 Kx = Hx = Cu 11(7, 8, 9, 10) \{ ( 0, 11)) Kn III di8 11(1, 7, 9, 10)\{(0, 11)} K, V 9 10 l l n 2 b I 11 o El6 1 1 (1, 6, 7, 8 ) \{(7, 8 ), (9, 10)) K, I 0 Fu 11(1, 7, 8)\{(7, 8), (9, 10), (0, 11)) K, j 4 V 7 8 o v

112 New Graph Split K /H K/H Type k / h KOh / ^ 6 ^ 10 V o 3 5 ill7 D5 6(2, 5)\((7, 8)} 1 ^~5 6 o 3 ^ 2 6 ^ i 3 i O Hx = 7 8 $ D12 5(U, 7, 8)\{(7, 8)} 1-^*5 7 n Q Kx = 3 U 2 6 ^ i O ' e19 5(3, ^,7)\((l, 7), (6, 8)J Kx ii H1 I E20 5(^,6, 7) 1 ^ 5 II & \ ( ( l, 8 ), (6, 7)} 3 it- 2 H1 <S> Fx 5(3, k, 6) \{ (l, 2), (3, 10, (7, 8)} I <S> i d>

113 101 New Graph S p lit K /H / v. r v K /H Type K /H k Hh / irnsr C10 6(3, 7, 9) 8 9 Kx = ^ = k 6 II c 6(1, 3, 5)V(0, 6)} Kx I H1 Dl8 6(1, 5, 7 ) \{ ( o,6 ) } ^ 8 9^2 V l l k 6 7 >11 V l o El6 10(7, 8 ) \{ ( 7, 8), (9, l l ) ) Kx I 0 e^2 10(7, 9)\((7, 6), (6, 9), (8, 11)} K, H I 0

114 102 New Graph S p lit K / H r*j,rw> K / H Type K / H KflH/ Dl8 6(7, 11) \ {(8, 9)) II / Oj VI O H = * E36 2(1, 5)V (l, 5), (3,M) K, Ek2 2(1, 3)\((1*, 5), (3, 8), (1,6)) K, I 0 Cl i 6(1, J*)\f(o, 6)) 2 i 6 h i Kj_ = 3 5 ^ 12 9 V H1 =

115 New Graph Split K /H Type k Hh / K /H K/H Kfl H 103 Dl8 6(1, 8)\{(0,6)} K, 2 1 $ t v ,i v o *4 \ o 3(1, 5)V (1,5), (2,4)} K. I 0 \ 2 3d, 4 )\((i, 6 ), (6, 4), (2, 5)} K, I 0 11 \ 2 11(8, 9)\((8, 9), (7, 10) n D,

116 New Graph Split K/H / v, / v K /H 1C* Type K 0 H / K /H K H H 1 (7, 8, 9 ) \f ( l, 0)} kl = l ^ = I E, 1(2, U, 6, 7)V(3, 7)} VI 0 E,8 1(2, *, 7, 8)\{(3, 9)3 K, 8? VI F1 i(u, 7, 8)\{(3, M, (3, 9)3 Kx D, H 5 0V o E2 2 ( 3, W ) \( ( 3,I O ) } ^ IV *Exception Oj0 ^ ""

117 New Type K 0 H / Graph Split K /H K / h K /H KflH E u 1(3, 7)V(2, 10)) M ta i 0 V K l = w <L * h A % 6 4 *? 0 71 d ^ = E23 1(2, 3, 10)\{(0, 7)} Kx * ^ Qs 3 J2. VI 10" 9 81 E32 1(2, 3, 5 )\{ (2, 3)} ^ 9? ^ 2 71 *<* 8>6 l 10 Vi Hx. Fu 1(2, 9 ) \ ( ( 3, o), (3, 10)) Kx Hi v. v 5 71 * 6* 9 ^ 0 F5 1(2, 3, 9 ) \ f ( l, 9), Kx 3A j ^ V 71 ' d ( 2,3 ) ) 2 0 \ i i 6 F1q 1(2, 3, 9 )\{ (l, 3), (2, 10)) Kx II $

118 106 New Graph S p lit K /H K /H Type k Hh / k / h k H h D5 1(2, k, 5) 1 2 y K± = b 5 3 o O ^ = K, O o D- 3(1, 5, 7) K, o 0 d8 1(2, 3, 5 ) K, o 6 E. 1(U, 5,6)\((U, 5 )} K, H. I o Vs E_ 1 (2, 3, 6)\{(U, 5)} K, I 0 0 /

119 Split K /H Type khh/ k / h k / h k Dh 3(1,1*, 5)\((1, 2)) 2 - ^ - 3 ^. U 5 l o l 5 3(1, 7, 8)\{(1, 2), (U, 5)} Kx i O 4 / o i t e r 3(2, U, 5) ) 2 Hj_ = c 2(1, 3 ) \ ( ( ^ 5)} K, I o 0 3(2, k, 8) \{ ( 2, K), (1, 5 )} ^ I O o 3(5, 6, 8)\{ (2, U), (1, 5 )} II O

120 D5 108 A «I Tv. New _ ^ Type k Hh / Graph S p lit K /H K /H K /H KflH Eg 6(U, 5)\{(U, 5)) 1"S~V i6 H = Eu 1(6, 2) \ {(If, 5)} Kx Hj^ Eig 1(2, 9)V(3, 7)) lj ^3^ K2 = U H VI 1 6*v E20 3(2, If, 7)\{(1, 9)} Kx II O H ,. 7 VI ^ 1 ^ 4 V F1 i(lf, 5 )\{(1f, 5), (3, 7)) Kx I o II O 7t /5 H = 6 9 2

121 109 New Graph Split K/H, / v k / h Type K /H "! / k nh F_ 1(5, 6)\{(3,»0, (5, 6)} K1 F3 3(7, 5)\((1, (5, 6)} K, I II O o Fu 1(2, 5)\C(3, 5), (^, 6)} K^ I n O O Bu 3(5,6) 7 1' -3. >6 Kx = h 5 2 H /fc. DlU 2(1, 5, 6) K, o H2 = E1q i (U,5,8)\{(U,5)J K, I 0 H. /

122 New Graph Split K/H Type K n H / <v. rs*l k / h k / h k H h En l(u, 5, 7 ) \ { ( ^ 5)) 6 2 * 3 o H n / Eig 1(2, U, 5 ) \( (2, 6)} i 6 A 9 5 u '2 i o II / Fx 1(2, 7, 8)\((2, 6), (K 5)) 1 y \ o r3 1(U, 7, 8) \{ ( 2, 5), (3,^)} Kx I 0. II o?5 1(2, k, 8 )\{ (2, fc), (3, 5)} ^ o

123 New ^ Type K (1H/ Graph Split K /H K /H K /H KflH I l l S10 xu 5 ( ^ 9) 3 - S i 9 i d K1 = 1 ^ 2 ^ = 1 7 = 3,6 8, 0 ^ h s d1]l 5(k, 6) k± i & Hx Dlk 2(3, K, 9) Kx a. H-L d D15 2(1, 3, 7) Kx i d Hx d E10 1(3,U,8)\((3,M) Kx i d H2 = l 9 7 T6 E12 1(2, 5, 8 )\{ (3,u )} Kx i (5 H2 y e 20 1(2, 3, ^)\{(2, 7)} K± II CL H1 3*M 5 8 ^

124 New Type K fl H / Graph Split K /H K /H K/H KHh 112 E23 2(1, 3, U)\C(1, 5)) 1 '2^ 2>9 1 ^ h E ^ 2(1, ^ 7 )\{(1, *0) K1 I ^ i i >1 >5 H3 = E25 2(1, U, 9)\{(1, U)} 3 ^ 5 ^ 8 I ^ h i? H3 X E2q 1(2, 3, 5 )\((2, 3)) >6 VI ^ l ^ F-L 2(3, 7, 9)V(l,»0, (3, 6)} K1 I H2 0 F2 1(2, h, 8 )\((2, U), <1 4?'>r (2, 3 )) K2 = ? ^? 3 = 9 6 l o

125 113 New Graph Split K /H K/H Type K/H KDh/ KflH F3 1 ( ^,5, 8 )\{ (2, 3 ), (**, 5)) Kx I 6 H 6?k 1(3, h, 5)\((2, 3), (l, Kr (X. M 87 3 U 6 1 VI 6 F5 2(1, 7, 9)V(l, 5), (3, 10) Kx I C 0 d Dlif 5(1, 6, 7) 3 - ^ 6 t f K1 = 7 V U Hx = E5 7(^, 5 ) \ ( ( 6, 9)J K, I 6 0 y E? 6(5, M \{ ( 3, 7)} K, I 0

126 New Type K,fiH/ Graph Split K /H K/H K /H K H H llu E2q 5(2, 3, 6)\((M 7)) Kx I 4. < ^ Hg = 9 J 1 2 Fx M l, 2, 7)\((5, 7)5 Kx I Hx O F Ml, 2,7)V (3, 5), A ^ 5 8 r (5,7 )} 39 ^ * 1 2.?,? 0 97 l 7 ' ^ 6 F3 k(6, 7 ) \ f ( 3, 5), (6, 7)) ^ i d h 8 V 5 0 d 1 2 * 3 G M l, 2 ) \ ( ( 3, 5), ^ 37,5 8 ^ ( 0,7 ), (5, 6)} 69 " " l 2 \79,. 5 8< 5,7 71 ^ ^ 0 6 D9

127 New Graph Split K /H ~ TyPe KHh/ k / h k / h k Hh Eu 6(5, 7 ) \ ( ( 1, 8)) K, = 6 U l 2 U , H1 = 1 3 v VI! E26 1(2, 8, 10)\((^, 7)} K, VI / II E27 1(7, 8, 9)M(^, 9)3 K, II 1 y2 10 VI F2 1 (2, 9, 10)\{ (^, 5 ), (6, 5), (5, 3)} K. 0 l)-v > f VI II 10 3 ElU 2(3, 9, 10)\f(9, 10)} 3 1 ^ -7 Kx = 5 ^ 0 2 II

128 New Graph Split K/H Type ns / k / h k / h k Dh E15 2(1, 3, 6)\{(9, 10) 8 0 ^ : 2 IV M 7 0- ^ M 7 e 23 2 ( 1, 9, i o ) \ { ( i, 1 )} Kx k: e32 2(1, 6, 9)\((i, 9)) K, VI E 1(2, 1, 10)\{(2, 10)} Kx II us- 1^ y6 VI ls5 8 l F 1(1, 8, 9 )\{(2, 10), 8 ~ (7,9)) 6 5 " ^ 3 1 f ' «^ s F 1(2, 1, 9)\{(2, 9), II a (l,*)) V f e

129 New Graph Split K /H Type K fl H / K /H K /H KfiH 117 F9 1(8, 9, 10)\{(1, 9), (2, 10)} II b VI 1 V 7 6^ 0 G 2(1, 3, 9 ) \ ( ( l, 9), (2, 9), (l, 10)} l o V 5 II d 3 5 ^ 2 0^ 1 11 Dl6 1(2, b, 10) K l = ^ = 8 1 6,9 i 3 n 'X 1 1 X e 23 2 (1, 9, 10) \ ( ( 1, *0) K, X. 9,o 8 VI 6y*s ti X e 32 1(2, b, 9 ) \ ( (2, 9 )} K, 0, 'l6,b vi

130 New ^ Type khh / Graph Split K /H K /H K /H k Hh F l(u,8, 9)\((6, 9), ^ d 5 6* ic>x (2,1 0 )} 5""^ 5 ^ VI 7 34 o ^ ^ 10 8* bj/ 8 6-^ O F,. u l(b, 9, 10)\{(2, 9), Kx * ^ (9, 10)} H 1 «i>1 0 1, n ^ F5 1(2, 4, 8)\((9, 10), 01, 6 V 53 I (2,7)} 2 ^ 8 > V W * *10 1 F10 1 (2, 8, 1 0 )\((1, 1 0 ), 7 * 1 9 v 3 <2 (2,9)} 2^5 - ^ VI ( i s ^ " G 1(2, 8, 1 0 )\((2, 10), (2, 9), (1, 0)} 7 i J o S' ^2

131 New ^ ^ Type K H H / Graph Split K /H K / h K /H k Hh E;li M 2, 3 ) \{ ( 2, 3)) I 6 Ki = o Y \ = e 19 1(2, 3,^ ) \ ( ( 5, 8)) 2 ^, 3 6 l-* -6 7 ^ HT - 6 ^v8 0 v i 1 *>< E26 1(2, 3, 9 )\((2, 3 )) Ki I 6 E27 i(u, 7, 9)\{(2, 3)} Kx I ( f H 0 / Px 1(3, 7, 9 )\((2, ^ ), (5, 6)} Kx I cr H VI ( f 7 9S S & 3 4

132 120 New Graph Split K /H K /H Type K /H KHh / khh F2 1(3, 7, 9)V(2,10, (3,6)} Kx I 6 0 F3 8 (7, 9 ) \ { ( l, 3), (2, b)} Kx i 6 H. i i 6 F4 1(3, M ) \ ( ( 2, 8), (3,M) 3 < * i 6 XX ^ 'i?ij K Dl8 6(3, 5) Kx = 7 10 T s 3 5 H1 = 2 k 6 e13 7(9, 10) \ ((9, 10)} I O F 7(9, 8 ) \( ( 9, 8), (6, 10)} ^ I O ^ II O

133 New Graph Split Type KHh/ K/H K/H K/H KHh 121 G 6(3, 7, 9)V(7, 9), (8, 10),'(0, 6 )} K 10 V b 8 * \l e lb Dl6 3(^, 10) I - 2-3y, Kx = 10 9 ^ i 6 l 3 76c r^4 J10 Hx = >6* D19 1(2, 5, 10) K, i i d 1 ^ 3 ^ ) 6»4 H2 = i i a : E12 10(1, 3)\((5, 6)) K, H_ 5(1, M \( ( 9, 10)) K, i a 0

134 New Type KHh/ Graph Split K /H K /H K /H Kl"lH E20 1(2, 9, 10)\{(2, 7)) Kx II H2 3 } / 6 VI E3q 1 (2, 9, 10) \{ ( 9, 1 0 )} l5 r^ 3 I & 10 9-g I ^ H3 = E31 1 (8, 9, 10) \ {(9, 1 0 )} K± I 0 1 5> 7 / h4 = 6 ^ 2 8 E32 3(2, 1 0 ) \( ( 1, 10)} K± I 0 Ttk / F 1(2, 5, 9 )\{ (7, 8), 1 6 3V, m *7 >10 (9, 10)} K2 = J^nS?*6 3- ^ 10 VI 3{ X *9 2<»( H = VC o F2 1(2, 8, 10)\((3, 9), - (3,1 0 )} VI 9 l*8 5

135 Type K 0 H / Split K /H K /H K /H KHh 9(1, 3 ) \ ( ( 1, 2), (5, 10)} Kx < r 6 1(5, 8, 9 )\((6, 9), Kx I (3, 1 0 )} H3 ct ct 1(5, 8, 1 0 ) \( ( l, 5), 11 (9, 10)) ^, 1 o 2 10l 6 VI 74x * 3 2-r >? ^ (5, 9, 1 0 )\((3, 9), 9 l ^ 3 ^ 7 H (1,1 0 ), 0^5)} 6 ^ V ^ -1 0 VI O+NjS + b 5 ^ 3 V 9 ^ 2 0 ^ 9 5 &

136 New Type K H H / Graph Split K /H K /H K/H 12b E ^ 10(2, 6)\f(l, 9)) Kx I & H2 = E 5(U, 9)\{(l, 10)) Kx I ^2. 1(2, 8, 9 )\{(6, 10), {(3, 9)) Kx I Cl h 2 II 0 F2 1(2, 8, 9)\{(2, 10), 9 ^ 8 23 II <Z (3, 9)} * b VI» w a 9 o 0 F 1(2, 5} 9 )\((1, 0 ), /? 4-f )10 (5,9)) 3 l X V9-10 VI OE ^10*3 4 f / i r 7 X 1, F9 1(5, 8, 9 )\((5, 9), (2, 10)} I cr 0 cr

137 125 New Graph Split K/H Type K H H / K /H K /H Knff 1(5, 9, 10)\{(1, 9), (2, 10), (b, 5 )) l VI 16 j f = i f 2 1(5, 9, 10) \{ (6, 11), (7, 10)) «2>»10. ^8 K = s VI 6? 10 9 A i'l l o F3 1(5, 9, 10)\{(3, 11), (7, 10)) K2 = c & 5 9 ^ 3 3 A0 10 VI = a 34< J 2 \ f 5 1(2, 9, u ) \ ( ( 3, n ), K <T (7, 10)} H_ 1 10 VI o 3 2 ^ f 9 1 (9, 10, n )\f(i, 0 ), (10, 11)) <L * 1, 7 10I VVf'VJltZ ,, 3 1 VI <Zi 5o ^ 9 2 o

138 New Graph Split K /H Type KHH/ K/H K/H itnh 126 F1q 1(2, 9, 10)\{(2, 10), (10, 11)) K. A f ^ 3 IV ( G 1(5, 9, 1 0 )\((2, 8), (6, 11), (1, 10)} K 2 5 \ 9 11 J 4 * IV Q D. IT E20 1(2, 5 ) \ ( ( 7, 6)} B 2 -*-7 I ' V i 2, 3 8 ^ 18 e36 10(9, 8)\C(9, 8)3 10 ^ 2 Kx = V O 3 5 l j i * ^ = /

139 127 New Graph Split K/H k/h Type K/H khh/ khh Pu 10(9,. 11) \ {(9, 11), (7, 8 )) K, I 0 O D 19 F3 11(1, 3)\((2, 3), (5,6)) 3T^ V K, = 11 '5-t-lo 8 ii 10,1 4+ / < 6/ 2 ^ = f 9 1(2, l l ) \ { ( 6, 10), K, I * (3, 11)) c r Fll+ 11(3, 10) \ {(3, 10), I $ (1, 5)) o ' 1 E, II Kx = 7 9 H = 11 2 U

140 New Graph Split k / h Type K /H K /H KHh/ khh EU2 11(1, 3,5 ) \f ( 0, 11)} k. I F? 11(1, 3,6, 1 0 )\{ (0, 11)} Kl a5! V ' n 7 9 % V 2 U l l 10 O Fi ;l 11(1, 3,6 ) \{ ( 0, ll) } K, n i / 2fo 9 7 V f t 0 2 ^ life V o ^" E1? 1(8, 10, 11) 1 I6 3P2 Kx = 11 if5 8 9 o 1 V % E± = s o E 3 q 1 ( 2, 6, 1 0 ) l ji 6 8 1? '< J

141 129 New Graph Split ~ ~ Type K /H K /H K /H K flh / itnff P10 1 (2, h, 1 1 )\{ (0, 8 )} K l0 0 b 11 VI o E, 8(1, k, 7) E,, ^ V. E? 8(7, 9) K± = H1 = 3 6 7' >8 E9 5(1, ^ 7 ) K, V*-

142 Split k / h /v i, rw/ K /H Type K/H KHh / A, A /V ' k Hh 2(1, 3, 7) k. V*r 2(3, ^,6 ) \ { ( 5,^ ) } K, 8 5> VI c r 2(1, 5) ^ = ) / Hx = ^ 2(1, 9) K, 1(2, b, 7) K 5(2,» 0 \ { ( 1, 6 )} K, ( * 8 2 b 0 VI

143 New Graph Split K /H Type K H H / K /H K /H KflH F2 5(2, 7, 8)\{(1, 2)} K, 9 1, VI h A 5'29 U 6 o d~v m T \ E 1(2, 7, 5) = l ^ 1 H = / F2 1(2, 5, 1 0 )\((3, 7)} Kx VI ^ o F6 1(7, 8, 10)\{(3, 7)3 Kx 3, 0 6 VI / l 8 2 5

144 New Graph Split K/H K /H Type K /H 132 khh/ ns? ElU 5(1, 1, 6) 2 5 1?< V ' F0 5(1, 7 ) \( (2, 7)} *5 6 VI CT F- 5(1, 7) \ ( ( 2, U)} VI <f J2 ^ 0 9 j \ F2 1(2, 7, 10)\{(3, 8)} 5 ^ 5 U VI / K1 = / 3 1 n O ^ = F3 1(2, b, 5 )\{ (2, 3)} Kx VI O *7 i 7 ** H2 =

145 133 New Graph Split K / H Tyj>e s j s / K /H K /H k Oh l(2f, 7, 10)\{(3, 5)) K, n O E29 2(1, 3, 7) ^ = 1 U 7 V > Hl = -,2 3 K K, VI l b o9 G 2(1, U, 7 )\((2, 7), (5, 7), (8, 7)} K, 8 s.5 2 VI b E. 10

146 Split K /H K /H Type K /H KH h / KflH 2 (3, 7) v K1 = 1 7 ^ 5 9 2, >8 Y>0 Hx = (1, U, 7) K, H 5(1, h, 3) K, V ' 5(3, 6 ) \ C(3, 9)) )2 6 h 7 Y >9 H2 = VI (1, 7 )\C (3, 5)3 Y',8 Q VI <S 5(U, 6, 7 ) \{ ( 7, 8 )) ; * 8 *2 ^ V 3 6

147 135 l l New Graph S p lit K /H Type K D H / K /H K /H k Dh E29 1(2, 5, 7) *2 Kx = H1 = E30 1(7, 8, 9) K, H. 1(5, 7, 9)V(3, 4)} K, >"<,4 l 8-t/l V o2 7 5 VI i i (5 F3 4(5, 7)\{(1, 8) K, VI 6 \ M3, 5)\((1, 7)} K, VI 6

148 136 New _ ^ Type k Dh / Graph S p lit K /H K /H K /H k Hh ElU 5(6, 10) 1 _ bn 10 ^ = Y3 H1 = E H i, 8) H E 1(5, 7, 8) K± / E3k 1(2, 5, 9) Kx ^ / F1 1(2, 7, 9 )\((3, 9)) K n 1 Y F3 1(2, 7, 8 )\{(U, 8 )) K± A \ 4 10 ^ / II 0 H1

149 S p lit k / h 137 Type KHh/ k/ h k/ h icnsf 5(k, 10) \ { ( 1, 9)} K. VI II < s 1(5, 7, 9 )\{ (1, 9)) K, \ 2 IV?t ^ II 0 K 5, 9, 8 )\{ (4, 5)} 0 J13 H (6, 7, 9) 5 3 i, o K± = 2 b 11 II / (3, 5, 6 ) \{ ( 6, 10)}

150 New Graph Split K /H K/H Type K /H 138 K flh / k Oh G 11(5, 6, 7 ) \ {(11, 12), (9, 1 0 )) V 1 V 2 If- 11 5J VI 4 3 ^ 7 6 E lb E1? 9(1, M K, = 7. 1! 9 6#\8 1 b 2 = V" e 37 11( 1, 2, U) K, 0 F3 11(1, 2, ^ ) \ { ( 5, 8)) K, 10,6 11 VI b " 0 3 F5 11(3, b, 5)\t(b, 8 )) K, 1 1 / 1 2 VI II (5

151 S p lit k / h Type KHH/ k/ h k/ h Knff 9(1, 10) \ ((5, 11)) K, i i 2o 6 K 10 vi (1, 2, 3 )\{(2, 9 )) K,,<f VI \ < l o ' b 0 i i CL J15 3[ 2(1, 6) 3 K1 = 8 U 2 11 H1 = H '* 21 1( 2, 10) K, l ( ^ 8, 9)M (7, 9)) K, / 1 ^ VI (2, U, 9 )\{ (5, 10)) Ka h. 11 8o^ 2 9 vi O /

152 i4 o New Graph S p lit K /H Type K 0 H / K /H K /H {CHS F9 1(2, 4, 9)V(2, 11)} K, II F10 1(2, 9, 1 0 )\t(2, 6)} K, t VI 0 1(2, 1, 9 ) \ ( ( l, 2)} II VI (5 G 2(1, 11)\((1, 8), (1, 4)} K, VI / II 16 E^0 12(3, 6, 9) = > II H1 =

153 lu i New Graph S p lit K /H Type KOH/ k/ h k/ h khh E^2 12(7, 9,1 1 ) K, I Fu 12(3, 7, 11) K, ^ 2 6 j 8 V 5 l 3^ V # J17.9 V F U ( l, 2, 5 ) \ [ ( >., 8 ) ) 1 ^ 3 27 * K ^ W 01 ^ = F1q 11(1, 2, 3 ) \( ( 2, 12)} Kx ^ H1 6, o ^ U Fllt 11(2, 3,1 0 M ( *,1 1 ) ) ^ - H i *

154 New Graph S p lit K /H Type khh/ k / h k / h k n h e 21 1(2, 8) c f 11 I f E 19 \ 8 ] 3 i E23 3(2, 8) 6 ^ 3 4 K1 = G 1 \ 6 F2 6(7, 8)\{(7, 8)} K, I 0 o G ' 3(2, U)\{(6, 9), (7, 8)} K, I <S> G

155 J20 1^3 *^7 8^ New Graph Split K / H ^. rw K / H Type k Hh / k / h k Hh E23 1(8, 9) i <S> Kl = i i s i/ K, I 8 f * / M3, 5)V(1, 9), (3, 5)) K. 8, 2 6 I 0 E 21 32V 1 ( 2,5 ) 5 fa 7 K1 = l / /

156 Ikk New Graph S p lit K /H k / h Type K /H K H H / g w E2q 7(^, 6) K. K VI 2 * ^ o P5 5(6, 8) \ { ( 1, 7)) K, VI )4 2 /+ o 6 8 o E25 9(1, h) ^ = e ^ = k E26 K 7, 9) K, II

157 S p lit K /H Type KflH/ k/ h k/ h koh 3(b, 10)\{(U, 10)} b 10 3d y ^ >? ^ 2 U IV u H V 4 f (5, 1 0 )\{ (5, 10), (*, 8)} p * : i i i 73.? 1; 5 9 VI e 2b 1(3, 6) i * 3 6 K, = 25 ^ i i**a> t (^, 7) K, H1 = ^ 4 Y 8(1;, 9 ) \ ( ( l, 2)} K, VI < f

158 1U6 New Graph Split K/H K /H K /H k ^ h / KHh l ( 6, 8 ) \ ( ( *, 8 ) ) K, V 1 ^ 6 8 X o F5 3(7, 1 0 ) \ { ( ^ 8)} < VI 1 10fv F9 3(7, 1 0 ) \( ( 1, 6)} Kn 3»*J+ 1 7 t 1 0 \t VI t E 25 E30 6(1, 5) 1 9 5, K = s 8 / 3 E1 = E33 9 (2, U) Kx H-L E35 1(2, 6) II ^

159 Split K/H K /H Type k / h k Hh / {cnff 6(5, 9)V(2, 9)) K 1, VI <S 9(2, 1 0 ) \ [ ( l, 10)} K, 9 ^, 6, VI V? t / 6 j26 5(9, 10) l 6 c 3 K1 = 2 V<k ^ = 7 5 ^ ^ 5 (^,1 0 )\{(3, 10)} Kn \

160 New Graph Split K/H k / h Type K/H 148 k Hh / Knsf E30 6(2, 5) K, F2 5(6, 9 ) \ ( ( 3, 10)} K, V ' 5, 1 8 VI 0 e y i 9 Fj 5(6, 9 ) V ( 3, *»)) K, VI 41 b i X (7, 9) \ ( ( 6, 7)) K, VI V- Hn (L G 6(2, 7)V(4, l), (4, 5 ), ( ^ 3 ) ) 6 3 o l 9 VI 2 7 ^ 5 08 VC j28 e32 1(2, 5) Kx = ^ >4 H = II

161 New Type khh/ Graph Split K / H K /H K /H KriH 2(1, 10) \ ((3, 9)) K± ^ Hi.9 1 «; 10b VI s5 9 2 h 7 cc P5 6(7, 9 ) \( ( 3, 9)) Kx ^ Hi 0 U F9 2(6, 10) \ [ ( l, 9)) Kx & Hn VI G 8 3 ^ 2 pio 1(2^ 9 ) \( ( 6, 9)) h n V 1U M 2-< 9y 5 10^ 7 G 6(2, 9 ) \{ ( 1, 9 ), (1, 2)) Kx

162 150 New Graph Split K /H / v. / *-' K / H Type K /H KHh / r n s f E37, % 3 > 2 11 ^ Kx = II V " h1 = V 56 3 > 2 r 8(1, 7)\((i*, 1 1 )) ^ ^ 8* x*5 V I V ' O 8 5 VI i F3 8(1, 7)V(3, 10)} K, II V> VI F1q 8(1, 10\{(2, 7)) Kx 8, VI V > 1 ^ 9o G 2(1, 7)\{(k, 8), (4, 11), Kx II ^ VI ^10 0 E30

163 New Graph Split K/H TJ? e j / K / H K / H KflH e 37 1( 9, 10) b,8 6 3 K = II \A- 3 V / f»2 H = U 11 1 II ^ F^ 5(b, 8 ) \ ( ( 2, 6 )) K, 5 3 ii7 VI t ' ^ o II ks: F9 5(6, 8 ) \ ( ( 7, 9)J 1 0 It V * II vi 67 ^ > ^ l F1q 1(8, 9) \ ((7, 9)) K, ^ 3 10 VI 6 J31 F3 5(1, U )\{ (6, 11)} 5 9? 2,0 11 ^ K1 = 1 3 V

164 152 New Graph Split K /H K /H Type K/H KHh/ khh F9 8(1, 9 ) \ ( ( 2, 7)) K, 0 6 VI 4 t/i 4s t v ^ <x J32 ^ 4 3^2!b8 11 ^ 6-^,1! <11 8 II Nf O 5 ^ 3 \ f 1 ^ = F^ 5(5, 10) \ ((3, 6)} K± \ H1 4, 7 10 VI d F 10(5, l l ) \ ( ( 3, 6 )) \ h n v i (f L 2-\/% F % b, 1 0 ) \ f ( l, 2)} K VI ^ 9 1, V r 10 0 ^ 11 0

165 New Graph S p lit K/H K /H Type K/H k n h / k H h F1q 5(6, 10)\{(2, 10)) K1 8 l n5 VI H1 II 6 G 10(2, 5 ) \ ( ( 1, 2), (1, 5)) K1 5,v0 t3 10 VI II e J33 E3t 6(1, 5) II / K± = / l 11 II V s =39 K 2,6 ) K, II / II / Fk 6 (5, 11) \ ((2, 9 )) K, II /

166 S p lit K /H Type khh/ K/H K/H K0 H 5(6, 1 0 )\{ (2, 9)3 K, ^S VI 6 E 3b 1( 2, 11) / 8,vio b n % h i H = / " 5(1, )+ ) \( ( 0, 5)) b 11 ^ 2 s i s 1 6*5 47 S i *J I. < & b(3, 9 ) \ { ( l, 2)) K, b 78, 5 VI 3t \ 9 1, 6 Nio o 5(fc, 1 0 ) \ { ( 1, 11), (*,7)3 K, 0, 8 2 VI /

167 J New Graph S p lit K /H Type KDh/ / ^ 7 / v ^ t k / h k / h khh e39 10(7, 9) 67., ^ >9 K = b ^ (1, 6 )\((2, 0) f t 5 6 M 11 I ' i V I ** F5 10(7, 1 1 )\{ (5, 11)} K. / VI (5 8-f ^ f 9 10(7, 1 1 )\{ (1, it-)} K, o, 7 / H VI / G 2(1, 10)\{(1, 10, (7, 10)} H K, H. ii *11 VI x 6 3W

168 S p lit K /H Type K H H / K /H K /H K0 H 7(11, 12) \ ((8, 9 )) 5 3 1*9 2 U V I V E. 37 \ 9 9(1, 1 0 )\{ (5, 12)) 7 _ U 9 K = l H 2 V ' 10, * Xl 11*^ *8 i l H = VI 9(2, 1 0 )\{ (2, 7)) K, 0, 8 c 11 VI i > f 5\ '38 1

169 S p lit k / h Type K fl H / K /H K /H Kfi H 1 (10, n ) \ { ( 3, *0) p l VI K l = i o ' W V i < T a. 2*1 ap I s H = 3? j ijk * * (10, n ) \ f ( i i, 12)} K. f i i 1(2, 11) \ {(2, 9), (8, 11)} K± h n i # 61 J39 ^ 7 J b 9 : ^ 4 3 p K 1(2, 9) K l. ^ r i i H l. s i s ^ 1 (2, 12) \ {(7, 10)} K - ^ 2 71 '

170 158 New Graph S p lit K /H K /H Type K /H K D h / k H h F1D 2(3, H)\{(2, 0)) 3f>Jo b v li2 III & V 7 ^ c G 2(1,11)\{(7,10), (1, U)} Kx \? i ^ VI II & ^ 0 12 I >0?, EU2 13(1, 5) K, III J > 3 = « = U 2 5 Fu 1 3 ( 1, 1 1 )\{ (1 3, 0 )) Ki i l l 7 10 V 1 a $ H1 3 6 V V E i+l

171 New Graph S p lit K /H /v K /H Type K /H K flh / k H h 159 1(2, 1 3 )\C (1, 0 )) ^,3\ * 2. ''V 4 " 8 7 ^ I 9 x 6 H* 10^ 5n 7* (9, 1 3 )\ ((7, 8 ), (3, 1 0 )) f t 9 13 ( 4f X t V ( 5 v? «2 2 v VI s F2 9 (3, 7) ^ 7 8 K = < r F3 9 (4, 7) K, 0 6 i i 6 F5 ' 8 ( 2,4 ) K, i i 6 n <r

172 l6o New Graph S p lit K /H /v. K /H Type K /H k D h / /v _ KDH G 8 (1, 2 ) \ { ( 7, 9 )J K. c r 2 VI 6 F8 7 ( 9,6 ) 9^ 7 8 =,05 ^ 2 1 K± =,M5 \ U e kx x = 39 5 ^ F10 8(1, 5) K, < r c c G 8 (1, 2 ) \ { ( 7, 9 )) K, O 1 k 5 VI 72k : 6 A 10 0

173 l6l New Graph S p lit K /H C V, ( V K /H Type K /H k H h /»v ' KflH F9 8 (2, *0 3 8 II K, 2 II G 8 (1, 2 ) \ { ( 7, 9 )) K, II < r i i * k 5 6 ^ 1 0 o VI e ^ > 7-4 s r F10 1 (5, 8) 8» /V K1 = U 2 10 E± = 4 " l 9 II II G 1 (2, 8 ) \ { ( 4, 1 0 )) K, II 0 8 w 2, VI & i d

174 162 New Graph S p lit K /H K /H Type K /H KHh / k Hh F9 9 (3, 5) io f Y 1 Kx = II o k 5 '? / < H1 = II ^ F10 9 (3, 7) K, O O ^ > 8 1 ^ 7 6 [ 1 (2, 9) 5 7 l j 9 K, = 8 6 V 10 II 3 10 V H1 = 2 9 U Fg 1(2, 6) K, II II II o < r F \ 5 _ n h o 9 6 K

175 163 New Graph S p lit K /H K /H Type K /H K H H / k H h 11(8, 10) 5 3 l f 9 = 2 k l l *1 *3* H = G 11(3, 1 0 ) \{ ( 0, 1 1 )) K, 5 3, 1 * V * 78 ^ 8 10 V e f 12 8 ( 1, U) 11 8 n K± =10f 6< ^ l k 5 ^ = 1 8 ^ G 8(1, 2 ) \ ( ( 7, 9 )} Kx H1 5 ^ VI n6^lo o

176 New Graph S p lit K /H Type K /H K /H K j H / k n h F13 7 (2, 11) K, 10 II <r 7 5 " 4 r 7 9 ^ = 10 " 6 11 II <2T Flk 7 (1, 11) K, II II c r G 7 (6, l l ) \ { ( 5, 10)} X t II CT 11 10, t < 5 ^ 7! 6 * VI (9 F 10 F12 7(1, 2) 9 10 II < 1 K?. < II <T G 7(1, 9) \ ( ( 2, 8 )) 11, i ' S i ^ II (r» a< o9 1 n VI e

177 These s ix graphs have a l l c u b ic v e r t i c e s a re so a re m inim al in 165 (I(P), <) Ei % 'lk ~ r

178 a p p e n d i x b This is a cross reference of the splittings by Type. TABLE k CROSS REFERENCE FOR SPLITTINGS Type I Al + Cl B6 - E5 C6 * El6 \ - e6 B13 A3 * C6 B7 " C3 V D5 D5 " E8 Dl ^ E A3 " C8 B9 * Dl^ C8 * El6 D6 " Fl Dl* + A3 ^ CU B10 ^ C9 C8 * eu2 D7 * D10» * + Ak + C10 Bn - d19 C9 * E36 D7 * E5 D15" \ * C11 Cl + El C9 * E42 D8 - E5 D1 5" 3 E_. P9 A5 " C11 c2 * Di cio - Ei,o D8 ->e7 D17 E2< b2 ->d? C2 + e6 C10 * EJ+2 d8 - f3 Dl8 E3f b3 * C2 CU * d15 cn - Eh2 D12 E11 D1 9" B5 " D7 C5 " E1 D3 E5 D12 "E27 E20" F 3 G b6 Dn C5 - E13 166

179 167 Type 1 A2 - El8 B2 ^ Bu \ \ - E21 B! E3 B3 " F1 B^ E5 C7 " F1 Type I, II A2 " B7 B, D7 * E12 Dl 4 - E3 l Bl ^ C7 B8 - C5 D5 ^ E11 D7 E23 DlU - E32 Bi " D3 B8 * D13 D * P 5 l D7 - ES* DlU * F5 B2 " D3 bio di 8 D7 " E25 D15 " Dl6 B2 * D8 C2 E8 D7 " Fl D15 * E35 C3 - D10 V F* D7 " F3 D15 - F1 B3 " D5 C3 " E28 D6 * D11 D8 " E28 D18 - F11 \ - b6 cb D12 d6 - Eio B8 F1 D19 * F9 \ - d6 C7 * D12 B6 * El l D12 - E26 '*19 * Fl^ b^ D7 C7 E19 D6 * El 9 D F 12 2 *19 * Ea Bl, * d8 7 * E 0 D6 * I 3 D > F 12 3 E -> F 19 2 BU D12 D3 D6 - F5 D12 ^ F^ e i 9 " G b6 " d 7 D3 * E19 D7 * DH D13 ^ F7 E20 _v E23 b6 * d8 d3. f i D7 * D15 \ b + d i6 S20 * PU b6 - e7 d^ F2 D7 - E10 Dl4 E30

180 Type 168 A1 -* A3 c5 - c9 E13 " E36 E27 * E29 A3 -* c6 * c10 \ - e 9 E1U E37 E3^ E39 C1 -V c6 C8 + C10 E6 * E13 E3 - E1m> P l -. P 3 C1 C8 Ei * E16 eio * E29 E2U E3 i f2 ->f8 C2 -V C5 Type, II B1 -V B2 D7 * D1U E10 * E31 E2^ " E32 Fl " F5 B1 b4 D8 "* DlU E11 * E29 E25 " E30 F F 2 10 B1 -V B5 Dl l - Dl6 E11 * E30 E25 " E33 F3 * F9 B2 -V B6 D13 * Dl8 E12 - \ b E25 " E35 *k * F10 B3 B8 di U Di9 E12 - E15 E26 E30 F -> F 5 9 \ B9 E2 * E17 E12 E33 E27 " E30 F5 * F10 B5 -V B9 E2 - E38 E12 E3U E28 " E32 F6 * F7 B8 -V Bio e3 * e4 El4 * E17 E29 " E37 F6 * F8 B9 Bn EU - Eio E15 * E17 E30 > E 37 F7 * F11 D3 -> D5 E5 " E7 E15 * E39 E32 * E38 D3 B6 E5 " E8 El8 + E21 E33 E37 F8 + F12 D3 D7 E5 * E12 E21 - E2L E33 + E39 F9 * F13 D3 -* D8 e 7 - El* E21 E28 E35 E39 F9 * % -V D13 E9 " E29 E2 2 " E25 E39 *' EUi F10 F12 b6 h k E10 - ElU E22 - E26 E! E2

181 Type I 169 A i - A5 \ - a 5 C6 cn cio cn E1 * EU2 A3 * A5 C1 * C11 C8 * CU Dl + El El6 \ 2 E, E35 " *k E39 * F10 E^ 0 - E^2 Ek l + F12 Type IV V D2 D 10 * El5 D2 + E2 E23 * F10 Type V B2 * C3 B2 - > Fi E12 - F10 Type V, B6 " B15 b6 - E20 B7 " D7 Type V, V A1 B8 A3 " Dl 8 B8 * G C6 - Fi l E1 - F11 A1 B10 Ah Bio Cl + D13 C8 + Dl8 ei 6 * Fn A1 - D13 B2 + C^ C1 * Dl 8 C10 Dl8 EUo + F11 Al " F6 B2 + E22 Cl " F7 D1 3 " G F? G A3 + B10 B8 * ei 9 C6 * di 8 E! * F7

182 Type VI 170 V E12 C3 + E25 D8 " F2 D3A * P1 D16 * f 9 V Fi c^ Fl D9 " EH Dl U " V P2 V P3 di + e 6 D u - v G D15 - P5 E15 * F10 C3 " E7 D2 - \ h Type VI, II a2 - c7 B -> E 9 27 D2 " F5 D10 - E32 D1U * p9 A2 - D3 B9 * F5 D F 2 10 D10 ^ E35 Dl ^ *k * E19 B11 P3 D5 * E19 D10 * P3 D15 * P2 B5 + C3 C2 - \ D5 * E20 D10 * P5 D1 5 " G b5 - cu C2 " D5 D7 * E20 D10 " p9 Dl6 - P2 B5 ^ El8 C3 * E23 D7 ** E28 D1 0 " G 3l6 " p3 B6 * E23 C3 " P5 D7 " F2 Dn - e23 D16 + p 5 b6 + E28 - G V F* Dn * E32 Dl6 * Fio B7 ^ E20 C5 * D13 Dg + G D11 * P3 di6 * G b 7 F1 C9 * D18 B9 " E26 D11 * *V E -* F 2 10 B9 * D15 Dl " E8 D9 * E27 Dl l - P5 E^ F1 B9 - E10 D2 * E23 D9 " F2 D11 * F10 V F1 B9 - E21 D2 - E32 D10 * Ell^ D12 - e i 9 E6 - F2 b9 E2U B2 - f1, D10 E23 \k " E20 e6 * f6

183 Type VI, (continuation) 171 E7 * p2 h h * P3 * P5 E2 9 * G E35 * G e 7 - p 3 V * P5 * p 9 E30 * F1 E36 * F11 E8 * P2 EH * P9 E25 * Fu E30 * P9 E37 * F9 E8 * F3 E,i P 1^ 10 E25 * F5 E30 * P10 E37 * P10 e 8 * pfc E15 * P 3 E31 * P3 E38 " P9 V F 2 E15 * P5 E27 * P2 E31 * P9 E38 * Fll. e9 - G E15 * P9 E -> F 27 3 E32 * e38 + G E10 - F1 E -* F i 15 1^ E27 * plt E32 * P5 E39 - P9 E10 * p3 E1 5 * G E > F 32 9 E39 * G E10 * F5 E -* F 17 9 E28 F* E32 F10 h i " G E -* F 11 2 E17 * P 10 E28 * P5 E3 2 ~ G F» G E11 * P3 E17 * PlA E28 * P 9 E33 * 7k E11 * \ E21 * Plt E28 * P 10 E33 * P5 f3 * g E12 * P1 E21 * P5 E2 8 * G E34 * p9 E12 P3 E23 * G E29 ** p2 e 3^ ''' G Fg* G 12 PU E2lt * P3 E29 * P3 E35 * P5 F9 * G E12 * P5 E2>t * Plt E29 * P10 e 35 * r 9 pi o * G E13 * P7

184 Type I, VI 172 A2 - Dl? B5 * E22 C2 * F1 A2 " E3 b6 - E21 \ * E25 B2 * E21 B7 - E21 D12 + F1 Type V, VI B^ F 1 V F1 B6 " F5 C5 * E19 cc 5 G b6 * F1 Bl l * F9 c 5 -> F2 E1 3 * G B5 * E20 B6 * F3 S - F5

185 APPENDIX C These are graphic representation of the generating sets of the partially ordered sets a line between G and in this work. H. If H *** S g \ y then there is v 173

186 fu^re <+3. (I(P'), ) (

187 j r*li

188 i*re 44. ( %CP)i ")

189 y \