Flexibility of Projective-Planar Embeddings

Transcription

1 Fleiility of Projetive-Plr Emeddigs Joh Mhrry Deprtmet of Mthemtis The Ohio Stte Uiversity Columus, OH USA Neil Roertso Deprtmet of Mthemtis The Ohio Stte Uiversity Columus, OH USA Diel Slilty Deprtmet of Mthemtis Wright Stte Uiversity Dyto, OH USA Septemer, 0 Vidy Sivrm Deprtmet of Mthemtis The Ohio Stte Uiversity Columus, OH USA Astrt Give two emeddigs σ d σ of leled oplr grph i the projetive ple, we give olletio of meuvers o projetive-plr emeddigs tht e used to tke σ to σ. Itrodutio Cosider leled oeted grph G with two ellulr emeddigs σ d σ i the projetive ple. I the se tht G is plr, it is show i [8] tht there re emeddigs ψ,..., ψ of G (with σ = ψ d σ = ψ ) suh tht ψ i+ is otied from ψ i y oe of list of give meuvers of grph emedded i the projetive ple. I this pper we solve the sme prolem for the se whe G is ot plr. This prolem hs previously ee osidered i [0] d [8]; however, oth oti errors. We will disuss these d relted results (whih re lso metioed i the et prgrph) i Setio. There re my results i the literture oerig the reemeddigs of grphs i vrious surfes; i prtiulr, results reltig the umer of reemeddigs to represettivity (i.e., fe width). The lssil result of Whitey [0] is tht y -oeted plr grph G hs uique emeddig i the ple. Roertso d Vitry [] showed tht for y orietle surfe S of geus g, y -oeted grph G emedded i Σ with represettivity t lest g + hs uique emeddig i tht surfe. Mohr [] d Seymour d Thoms [] lowered this oud to log(g)/ log(log(g)). Roertso, Zh, d Zho [] showed tht, other th the three emeddigs of C C i the torus, y grph with emeddig of represettivity t lest i the torus hs uique emeddig i the torus. Roertso d Mohr [] hve show tht for y surfe S, there is umer f(s) suh tht for y -oeted E-mil ddress: mhrry@mth.ohio-stte.edu E-mil ddress: roertso@mth.ohio-stte.edu E-mil ddress: vidy@mth.ohio-stte.edu E-mil ddress: diel.slilty@wright.edu

2 grph G, there re t most f(s) distit emeddigs of G i S with represettivity t lest. As orollries to our mi result (i Setio ) we will reprove the followig three relted results of Negmi d Vitry for projetive-plr emeddigs. First, for y -oeted grph with emeddig of represettivity t lest i the projetive ple, tht is the oly emeddig (Theorem.). Seod, other th K, for y -oeted grph with emeddig of represettivity i the projetive ple, tht is the oly emeddig (Theorem.). Lst, if G is -oeted d hs -represettive emeddig i the projetive ple, the the umer of emeddigs of G i the projetive ple is divisor of (Theorem.). Before we stte our mi result, we would lso like to metio reltioship etwee this prolem d prolem o siged grphs. If Σ d Σ re two siged grphs with the sme leled edge set, the whe does M(Σ ) = M(Σ )? (Here M(Σ i ) is the frme mtroid of the siged grph Σ i. See [] for itrodutio to siged grphs d their mtroids.) Sie the reltioship etwee differet represettios of the sme mtroid is very importt i mtroid theory, swer to this questio is desirle. I [] it is show tht if M(Σ ) is oeted d ot grphi, the M(Σ ) = M (G) for some ordiry grph G iff Σ d G re topologil duls i the projetive ple. So if M(Σ ) = M(Σ ) = M (G) is -oeted, the Whitey s -Isomorphism Theorem (see, for emple, [, Se..]) tells us tht G is the oly ordiry grph tht represets M (G). Thus the differee etwee Σ d Σ is just tht they re duls of two distit emeddigs of G i the projetive ple. Hee this prolem of siged-grph mtroid isomorphism otis the reemeddig prolem of projetive-plr grphs s speil se. I Setio, we will defie three opertios o grph emedded i the projetive ple: Q-Twists, P-Twists, d W-Twists. (Here Q, P, d W std for qudrilterl, Peterse, d Whitey.) Our result is the followig. Theorem.. Let G e oeted, oplr grph. If σ d σ re two emeddigs of G o the projetive ple, the there eists sequee of Q-Twists, P-Twists, d W-Twists tkig σ to σ. Our mi lemm to the proof of Theorem. (whih we sped the vst mjority of the pper provig) is Lemm.. Lemm.. Let G e -oeted, oplr grph. If σ d σ re two emeddigs of G o the projetive ple, the there eists sequee of Q-Twists d P-Twists tkig σ to σ. A turl pproh to provig Lemm. would e to fid topologil sugrph H of G with kow fleiility i the projetive ple d the emie the H-ridges d how they ehve uder the fleiility of H. A turl didte for H would e K, -sudivisio euse G is gurteed to hve it uless G = K. However, it seems to us tht suh pproh is ot fesile d so we strt with sugrph H tht is sudivisio of the Wger grph V 8. The Wger grph V 8 (lso lled the -rug Möius Ldder) is otied from 8-yle o verties v, v,..., v 8 y ddig four v i v i+ -hords. I y projetive-plr emeddig of V 8 its otgo must e emedded otrtily; with K, there is o suh yle tht is gurteed to e otrtile. This property of V 8 mkes proof of Lemm. strtig with V 8 -sudivisio trtle s the reder will see i Setio. Thus we split the proof of Lemm. ito two ses: where G otis V 8 -mior (Setio ) d where G is V 8 -free (Setio ). The V 8 -free se is filitted y Theorem. whih is result prove idepedetly y oth Kelms d Roertso ut remis upulished. Oe might sk whether oe ould eted the tehiques of Setio to grphs with K, -mior ut o V 8 -mior i order to void usig Theorem.. We oted tht suh pproh would tully just reprodue muh of the detils of proof of Theorem.. They lso give emples to show tht o suh oud eists depedig oly o the surfe for highly-oeted -represettive emeddigs.

3 Theorem.. Let G e iterlly -oeted grph with o V 8 -mior. The G elogs to oe of the followig fmilies:. Plr grphs. Sugrphs of doule wheels (i.e., there eist two verties, of G suh tht G \ {, } is yle). Grphs with -verte edge over (i.e., there eist four verties,,, d of G suh tht V (G) \ {,,, d} is edgeless). The lie grph of K,. Grphs with seve or fewer verties Twistig Opertios A Q-Twist is oe of the opertios desried i this prgrph. The full Q-Twist opertio higed t,,, d lthed t A, B is the opertio show i Figure for grph emedded i the projetive ple. Oe idetify d/or delete higes d lthes of the full Q-Twist to oti degeerte Q-Twist. Severl degeerte Q-Twists re show i Figure. For the rightmost degeerte Q-Twist, whe the light grey lok is sigle edge we ofte refer to this opertio s flippig edge. B A A B B A Figure. The Q-Twist opertio higed t,,, d lthed t A, B. B B= A=B== B== B A= A= = B B B= B= A=B== A=B== B== B== A= A= = B B Figure. B= A=B== B== A P-Twist is oe of the opertios desried i this prgrph. The full P-Twist is the opertio show i the first olum of Figure for grph emedded i the projetive ple. The seod d third olums of Figure re differet drwigs of the P-Twist. We refer these three drwigs, respetively, s the owtie, etrl, d petgol views. A degeerte P-Twist is otied from full P-Twist y otrtig trigulr pth or y otrtig oe side of trigulr pth. I Figure we show three typil degeerte P-Twists ll otied from the owtie view of the full P-Twist. The first is

4 otied y otrtig pthes,,8,, the seod is otied y otrtig pth, d the third is otied from the seod y otrtig the drk pthes to two edges s leled Figure The three views of the P-Twist ,8, 0,8 Figure It is worth otig tht the full P-Twist without the shdig i the trigles is the lie grph of the Peterse grph, ll it L(P ) where P is the Peterse grph. Oe hek tht there re etly two distit emeddigs of P. Now sie P is ui, eh emeddig of P eteds uiquely to emeddig of L(P ) d y emeddig of L(P ) omes from emeddig of P. Hee there re oly two emeddigs of L(P ) d these emeddigs re relted y the full P-Twist. Furthermore, the two emeddigs ot e relted y Q-Twist (or sequee of Q-Twists) euse i Q-Twist there re t most verties whose rottio of edges (up to reversl) is hged y the opertio. I the two emeddigs of the lie grph of the Peterse grph this hge i rottio ours t ll verties. Filly, W-Twist higed o verties A d B is the opertio show i Figure. 0 0,8, 0,8 A B A B Figure. Proof of Theorem. I this setio we prove Theorem. ssumig Lemm.. Our proof will proeed y idutio o V (G) + E(G). I the se se V (G) + E(G) = d so G = K or K, whih re oth - oeted. The result the follows y Lemm.. Suppose ow tht V (G) + E(G) > d G is oeted ut ot -oeted (the -oeted se follows y Lemm.).

5 Propositio.. If G is oeted, oplr, projetive plr, d ot -oeted, the G = H t P for t {, } where P is plr d H is ot. Proof. Give tht G is oeted ut ot -oeted, G = H t P for t {, }. Sie plrity is losed uder -sums d -sums, the without loss of geerlity H is ot plr. It must e tht P is plr euse otherwise G will oti oe of the twelve eluded miors for projetive plrity tht re ot -oeted. A proof for t = is evidet d proof for t = e foud i [, ]. Hee we ssume tht G = H t P for t {, } where P is plr d H is ot. Let {, y} e the verties of the seprtio d e e the y-edge of H d P. So if σ d σ re distit emeddigs of G i the projetive ple, these restrit to emeddigs σ i H d σ i P where the emeddigs of P re plr emeddigs iside disk. Hee, y idutio, there is sequee of Q-, P- d W-twists tht tkes σ H to σ H. Also y Whitey s theorem [, Thm..8], there is sequee of W-twists tht tkes σ P to σ P. These opertios o H d o P ll e performed idepedetly of eh other. Proof of Lemm. for grphs with V 8 -mior The -rug Möius ldder V is the grph otied from the yle o verte set {,,..., } y ddig (i, + i)-edge for eh i. Note tht V = K d V = K,. I this setio we prove Lemm. i the se tht G hs V -mior for some. Let ν 0 e the oil projetive-plr emeddig of V with fil -yle. Let ν i with i {,,..., } e the emeddig otied from ν 0 y flippig i the (i, i + )-hord. (Figure shows the emeddigs ν 0 d ν for V 8.) For, oe hek tht there is o emeddig of V i whih the -yle is ootrtile. Hee ν 0, ν,..., ν re ll of the emeddigs of V for. Figure. 8 8 Now let G e -oeted grph with two distit emeddigs o the projetive ple, σ d σ. Let H e V -sudivisio otied i G with mimum. Sie G is -oeted we ow rehoose H so tht it hs o lol H-ridges (see, e.g., [, Lemm..]). This implies tht y rh of H tht e hose to e sigle edge is hose s suh. Let γ i,j e the rh of H orrespodig to the (i, j)-edge of V d let C H e the yle i H orrespodig to the -yle of V. Let σ H d σ H e the restritios of the two emeddigs σ d σ to H. Without loss of geerlity, we split the prolem ito the followig five ses. I Cse, suppose σ H = σ H = ν 0. I Cse, suppose σ H = ν 0 d σ H = ν. I Cse, suppose σ H = σ H = ν. I Cse, suppose σ H = ν d σ H = ν. Filly, i Cse, suppose σ H = ν d σ H = ν k with k {, }. Before egiig our se lysis, we will desrie some geerl priiples tht we will use i ll (or most) of the ses. Two emeddigs of G i losed surfe S re the sme iff they hve the sme fil oudry wlks iff G is fied poitwise up to isotopy o S. Cosider y emeddig ψ of G i the projetive ple. Sie y emeddig of H G is -represettive, every fil oudry of H is yle i G. Now let A e fil oudry yle of H of legth l, let B,..., B t e the H-ridges of G tht re emedded iside of A, d let K e the grph K,l with degree- verties to e tthed to the verties of A. Sie G is -oeted, we get tht K A B B t is -oeted plr grph. As suh the fil

6 yles of K A B B t re uiquely determied. Thus emeddig ψ of G i the projetive ple is uiquely determied y whih fe of H give ridge is emedded i. Sie C H is otrtile i ll emeddigs of H, we sy tht H-ridge B is reemedded with respet to σ d σ if B is iside the disk regio of C H i etly oe of the emeddigs. Otherwise, we sy B is fied with respet to σ d σ. We ll B reemeddle whe there is emeddig σ of H B with σ H = σ H suh tht B is reemedded with respet to σ d σ. Evidetly y reemedded ridge is reemeddle d fied ridge my or my ot e reemeddle. Give σ d σ, let H e the sugrph of H with edges d iterior verties of the hords of C H tht re flipped reltive to σ d σ removed. We ll this the fied sugrph of H. Note tht i Cses d, H = H; i Cse, H is V -sudivisio with ; d i Cses d, H is V -sudivisio with. I Cse, σ H = σ H = ν d i ll the remiig ses, σ H = σ H = ν 0. So ow i Cses (euse ) the fe of H i whih give H-ridge B is emedded i σ k is uiquely determied y whether B is iterior or eterior to C H i σ k. Thus the emeddigs σ d σ whe restrited to H B re the sme (i.e., fied poitwise up to isotopy) iff B is fied ridge. I Cses d with, we get the sme result for H-ridges d H. I Cse with =, we do ot, priori, get this result for the fes of B i H. The oly time this might fil is whe B hs ll of its tthmets re o the γ, - d γ,8 -hords euse there re etly two fes i H tht re eterior to C H d B my e emedded i either oe. I this se, however, sie oe of the fes of H eterior to C H is lso fe of H, this ot hppe. So gi i Cse we get tht σ d σ restrited to H B re the sme iff B is fied ridge. I Cse with =, this possiility does ideed hppe (e.g., with the two emeddigs of the Peterse grph reltive to y V 8 -sudivisio). Therefore i ll ses sve Cse with =, the emeddigs σ d σ restrited to H d its fied H-ridges re the sme emeddigs. Tht is the differee etwee σ d σ is desried etly y whih H-ridges re reemedded. I Cse with =, we do ot use the terms fied d reemedded. Two H-ridges i G with tthmets o yle C re lled skewed with respet to C if they shre three ommo tthmets o C or hve pirs of ltertig tthmets o C. Also we sy tht B elogs to fe F of H i ν i if ll of its tthmets re o F. Note tht if two H-ridges B d B oth elog to F d re skewed with respet to F the oe of B d B must elog to some other fe s well. The si strtegy for eh se (eept Cse with = ) is to first idetify wht the reemeddle H-ridges re d the to defie sequee of Q-Twists tkig σ to σ. I this proof for grphs with V 8 -miors, P-Twists oly pper i Cse with =. Whe defiig potetil twist we must lwys verify tht there re o ridges (fied or otherwise) tht ostrut it. Cse : Note tht y H-ridge B is reemeddle iff ll of the tthmets of B re o γ i,i+ γ i+,i++ for some i {,,..., }. So the eh reemeddle ridge elogs to C H d the other fe of σ H = σ H with γ i,i+ γ i+,i++ o its oudry. Cll this ltter fe F i. If B is reemeddle, the either B is sigle edge with oe edpoit o γ i,i+ d the other o γ +i,+i+, B hs etly oe tthmet o either γ i,i+ or γ i+,i++ d t lest two tthmets o the other s o the left i Figure, or B hs ut-verte i its iterior s o the right i Figure. We will ll sigle-edge ridges I-type ridges, the seod kid V -type ridges, d the lst kid X-type ridges. For V -type ridges, let the sigulr tthmet e lled its pe.

7 B B B B B Figure. As V ws tke with miml, we get the followig restritios o these three reemeddle types of ridges. For I-type ridges, t lest oe tthmet must e rh verte of H. For V -type ridges, either the pe is rh verte of H or ll of the o-pe tthmets re rh verties of H. These will e lled V ed -type d V it -type for pe o rh verte d pe i the iterior of rh, respetively. For X-type ridges, they ot hve iterior tthmets o oth rhes. If B,..., B k re ll reemeddle, ll o-skewed with respet to C H, d ll iterior to C H i σ j the B,..., B k ll must elog to the sme F i. The restritios o the ridge strutures i the previous prgrph yield the followig possile ofigurtios for γ i,i+ γ i+,i++ B B k iside C H. If there is V it -type ridge i {B,..., B k }, the the remiig ridges i {B,..., B k } must ll e V ed -type d I-type d γ i,i+ γ i+,i++ B B k must e s o the left of Figure 8. If there is X-type ridge i {B,..., B k }, the the remiig ridges i {B,..., B k } must ll e V ed -type d I-type d γ i,i+ γ i+,i++ B B k must e s i the seod ofigurtio i Figure 8. If there is o V it -type or X-type ridge i {B,..., B k }, the γ i,i+ γ i+,i++ B B k hve oe of the two remiig types of ofigurtios i Figure 8. Figure 8. Cofigurtios of reemeddle o-skewed ridges iterior to C H A olletio F of reemeddle V -type d I-type ridges shrig the sme pe d emedded i the sme fe of H is lled f with pe. The other tthmets of the ridges i F must ll lie o the sme rh of H. We will ll the first d lst suh tthmets o this rh the etreme feet of F. So ow if B {B,..., B k } re ridges tht re reemedded from σ to σ, sie there ot e disjoit H-pths from γ i,i+ to γ i+,i++ withi the ridges of B, either B is sigle X-type ridge or f. As with V -type ridges, we further desrie fs s iterior fs or edpoit fs for whe the pe is i the iterior of rh of H or o rh verte of H. Iterior fs osist of t most oe V it -type ridge log with t most two I-type ridges. We do ot osider sigle I-type ridge s iterior f. Sie σ σ, there is some reemedded H-ridge. Either ll of the reemedded ridges elog to some F i or ot. Let the ltter possiility e Cse. d the former possiility e Cse.. Cse. Sie reemeddle ridges elogig to differet F i s re skewed with respet to C H, eh reemedded ridge elogs to oe of F i d F j for some i j. Let B i e the olletio of reemeddle ridges elogig to F i d B j e the olletio of reemeddle ridges elogig to F j. Let B i B i d B j B j e the ridges tht re tully reemedded. Sie ridges i B i re skewed o C H to ridges i B j, we hve without loss of geerlity, tht the ridges of B i re iterior to C H i σ, the

8 ridges of B j re iterior to C H i σ, d the ridges of B i \ B i d B j \ B j re ll eterior to C H. We ow go from σ to σ y first reemeddig eh ridge of B i idividully y degeerte Q-Twist. (We ot eessrily reemed B i ll t oe whe it is f s there my e reemeddle ut fied ridges tht lok this.) We the reemed eh ridge of B j idividully y sigle degeerte Q-Twist. Cse. Let B e the reemedle ridges elogig to F i d B B e the oes tht re tully reemedded. I Cse.. there is X-type ridge i B. I Cse.. sy there is o X-type ridge i B d there iterior f i B. I Cse.. sy there is o X-type d o iterior f i B. Cse.. If there is X-type ridge X B, the sy without loss of geerlity X is iterior to C H i σ. Thus the remiig ridges i B re ll eterior to C H i σ. Hee B \ X is either sigle X-type ridge or B \ X is f with pe. If B \ X is sigle X-type ridge, ll it X, the we go from σ to σ y oe full Q-Twist higed o the etreme tthmets of X X o γ i,i+ d γ i+,i++ d lthed t the ut verties i the iteriors of X d X. If B \ X is f with pe, the we go from σ to σ y oe Q-Twist higed o the etreme tthmets of B o γ i,i+ d γ i+,i++ d lthed t the ut verte i the iterior of X d t sve i emeddigs like the oe o the left i Figure where there re ridges i the f B \ X whose o-pe tthmets re o pth ot itersetig the pths etwee the etreme feet of X. Eh of these ridges must first e reemedded idividully y degeerte Q-Twists to oti emeddig σ show o the fr right of Figure. Let B B e the ridges remiig i F i i σ. We ow go from σ to σ (the seod emeddig i Figure ) y Q-Twist higed o the etreme feet of B X d lthed o d the ut verte of X. Figure. Cse..: The lower qudrilterl is the iterior of C H. Cse.. Let V e the reemedded iterior f. Without loss of geerlity, ssume tht V is iterior to C H i σ. If B = V, the we go from σ to σ y sigle degeerte Q-Twist. Otherwise, B \ V is iterior or ed f iside F i i σ. Let these e Cses... d... Cse... If there is lso iterior f, ll it N, i B \ V, the the possiilities for B i σ re show i the first three ofigurtios of Figure 0. We show the ofigurtios with V it ridges iluded, ut they tully eed ot e there or eed ot e reemedded; however they e dded d/or mde reemeded s there e o other types of H-ridges lokig this modifitio. We ow go from σ to σ y sigle Q-Twist higed o the etreme tthmets of V d N whe there is o fied V it -type ridge i V or N; otherwise, we go from σ to σ y two suh Q-Twists where the first Q-Twist reemeds the V it -type ridge(s) i V d N d the seod puts them k. Figure 0. Cse..: The lower qudrilterl is the iterior of C H. 8

9 Cse... Let N = B \ V. The pe of N is either o the sme rh of H s the pe of V or ot. I the ltter se, the ofigurtio for V d N is the fourth oe show i Figure 0. (Agi the V it -type ridge might ot e i V, ut e dded to it.) We ow go from σ to σ y sigle Q-Twists higed o the o-pe tthmets of V d lthed t the pies of V d N. No fied ridges lok this. I the former se, the ofigurtio for V d N is lst show i Figure 0. Let N r e the ridges of N whose etreme tthmets re oth t or to the right of the pe of V. We ow go from σ to σ y first performig Q-Twists to idividully reemed eh ridge i N r. Seod, for the remiig ridges of N \N r, there e o fied H-ridges with tthmets o the pth oetig the etreme tthmets of N \ N r. Hee we ow perform Q-Twist higed o the etreme tthmets of N \ N r d V d lthed o the pe verties of N d V. Cse.. Here B osists of oe or two ed fs. We ssume tht there re two s the detils for just oe re otied i the proof for two. Deote these two fs y N d N. We split this Cse ito the followig suses: i Cse..., oth fs shre the sme pe, sy o γ i,i+ ; i Cse..., the pies of the fs re the distit edpoits of γ i,i+ ; i Cse..., the pies of the two fs re o differet rhes. Cse... The f N j is otied i possily lrger f N j with pe whih might ilude reemedle ut fied ridges. Suh ridges re show i lk i Figure. Cosider two H-ridges B N d B N whose pths etwee their etreme feet o γ i+,i++ overlp i t lest edge. It must e tht B d B re oth fied or oth reemedded. Therefore, the pth γ i+,i++ deomposes ito supths P,..., P k (some possily of legth zero) with the followig properties: first, P m P m+ is sigle verte for eh m k ; seod, eh ridge i N N hs ll of its tthmets (side from ) o some i P l ; third, eh I-type ridge i N N the e ssiged to sigle P l otiig its o-pe tthmet suh tht ll of the ridges whih re ssiged to P l re either ll reemedded or ll fied; d fourth, if ll ridges ssiged to P l re fied, the ll ridges ssiged to P l+ re ll reemedded. Thus we tke σ to σ y sequee of k degeerte Q-twists, higed t edpoits of the P l s d lthed t. Figure. Cse...: Fied ridges re show i lk. Cse... Let t e the pe of N t. Without loss of geerlity, sy N is iterior to C H i σ. The first emeddig i Figure shows N d N i σ where the lower qudrilterl is iterior to C H. The seod emeddig i Figure shows N d N i σ. Give {j, k} = {, }, it my e the se tht there re ridges i N j tht hve o tthmet o γ i+,i++ tht is to the left of the rightmost tthmet of N k. If so, the ssume without loss of geerlity tht j = suh s wht is show i Figure. Let N N e the olletio of these ridges. Eh idividul ridge i N my e reemedded usig degeerte Q-Twist to oti emeddig σ show s the third emeddig i Figure. Now give {, y} = {, }, it my e the se tht there re ridges i N tht hve o tthmet o γ i+,i++ tht is to the right of the leftmost tthmet of N y. We reemed ll of the ridges i N (N \ N ) sve these y sigle Q-Twist (o fied ridge my lok this Q-Twist) to oti emeddig σ suh s wht is show s the fourth emeddig of Figure. Filly we go from σ to σ y performig idividul Q-Twists o eh of the remiig ridges of B.

10 Figure. Cse... Cse... Let t e the pe of N t. Beuse d re o differet rhes of H, the ridges of N re skewed to the ridges of N i oe of F i or C H d ot i the other. Without loss of geerlity, sy N is emedded i the skewed fe i σ. I Figure the skewed fe is the lower squre regio, the first d seod emeddigs re σ d σ. We my ow reemed eh ridge of N idividully y Q-Twist to oti the the third emeddig i Figure. We the reemed eh ridge of N idividully y Q-Twist to oti σ. Figure. Cse... Cse Suppose σ H = ν 0 d σ H = ν (See Figure ). Figure. Emeddigs of H i Cse There re three types of H-ridges tht re reemeddle. First, y H-ridge B with tthmets i the iterior of γ,+ must e reemedded d so ll of the other tthmets of B re o etly oe of γ,, γ,, γ,+, γ +,+. Let B, B, B, d B +, respetively, e the olletios of these ridges with tthmets o the iterior γ,+. Seod, H-ridges tht hve ll tthmets o γ, γ, or ll tthmets o γ,+ γ +,+. These types of ridges must e reemedded d we deote the olletios of these ridges y B d B +. Let B e the olletio of ll of these ridges i B z log with γ,+. Third, ridges with tthmets o oth pths γ, γ, d γ,+ γ +,+. Let A deote the olletio of these types of ridges. Rell tht for y olletio of reemedded H-ridges tht re ll iterior or ll eterior to C H, there must e verte whih prevets two verte-disjoit C H -pths i the olletio euse y two suh pths would ross if reemedded. Let α e the supth of γ, γ, etwee the etreme tthmets of B. Let β e the similrly defied supth of γ +,+ γ +,. Ay H-ridge i A eterior to C H i σ with tthmets produig 0

11 H-pth tht would ross some H-pth of B if it were drw eterior to C H must e reemedded ridge. Let A e the olletio of suh H-ridges. Now let H = H ( B) d we perform Q-twist higed o the edpoits of α d β i σ H d lthed t some verte γ,+. If this Q-Twist e eteded to ll of G i σ, the we oti emeddig σ for whih σ H = σ H = ν 0 d the we go from σ to σ y sequee of Q-Twists s i Cse. If we ot perform this Q-Twist o ll of G, the A. Let H = H ( A ), α eteds α to the etreme tthmets of A o γ, γ,, d β eteds β similrly. Ay H-ridge iterior to C H i σ with tthmets produig H-pth tht would ross some H-pth of (B A ) if it were drw iterior to C H must e reemedded ridge. Let A e the olletio of suh H-ridges. We ow perform Q-twist higed o the edpoits of α d β i σ H d lthed t verties d. If this Q-Twist e eteded to ll of G i σ, the we oti emeddig σ d the go to σ s i Cse. If ot, the A. Let H = H ( A ), α eteds α to the etreme tthmets of A o γ, γ,, d β eteds β similrly. Ay H-ridge eterior to C H i σ with tthmets produig H-pth tht would ross some H-pth of (B A A ) if it were drw eterior to C H must e reemedded ridge. Let A e the olletio of suh H-ridges. If A =, we fiish y etedig to ll of G Q-Twist o σ H higed o the edpoits of α d β d lthed t d d the refer to Cse. If A, the we iterte this proess gi. Of ourse, this proess must ed s G is fiite. Iside C H + Figure. A illustrtio of the itertive proess: the light grey ridges re B, the drk grey ridge is A, d the lk ridge is A. Cse Suppose σ H = σ H = ν. Cosider the fes of H s leled i Figure. T Figure. Emeddigs of H i Cse R S + + T T S R + + T The oly types of reemeddle ridges re those with ll tthmets o γ, γ,+ or ll tthmets o γ, γ +,+. Let S d R, respetfully, e the olletios of suh ridges tht re tully reemedded. If there is some reemedded ridge B R S whih otis H-pth with oth edpoits off of {v, v, v +, v }, the we my use this pth to rehoose H so tht we revert k to Cse. Thus the reemedded ridges of R S form oe or two fs (fs re desried i Cse ) with pies i {v, v, v +, v }. So ow ll of the reemeddig of G is hppeig withi -regio Möius strip with regios R S d T s i Cse.. with ll reemedded ridges formig fs with pies i {v, v, v +, v }. Whe S is set of sets, we use S to deote the uio of the sets i S.

12 The oly differee etwee our urret situtio d Cse.. is tht γ,+ uts oe regio of the strip d tht there my e fied H-ridges tthed to γ,+. However, γ,+ d y of these fied ridges ot iterfere with the reemeddig of the fs s desried i Cse... Thus we my tke σ to σ s i Cse... Cse Suppose σ H = ν d σ H = ν (see Figure ). Figure Prtitio the reemedded (ot just reemedle) ridges ito two sets B i d B out for those tht re iterior d eterior, respetively, to C H i σ. Note tht y H-ridge with tthmet o the iterior of γ,+ is i B i d y H-ridge with tthmet i the iterior of γ,+ is i B out. Suppose tht B is H-ridge with tthmet i the iterior of γ,+. The ll of the other tthmets of B re o etly oe of γ,, γ, γ,, γ,+, γ +,+ γ +,+. Let B,, B,,, B,, B,+,+, respetively, e the olletios of these ridges with tthmets o γ,+. Similrly ridges with tthmets i the iterior of γ,+ prtitio ito the followig four lsses: B,,, B,, B,+,, B,+. Let B e the olletio H-ridges with ll tthmets o γ, γ,. Let B + e the olletio of H-ridges with ll tthmets γ,+ γ +,+. Note B B + B out. Similrly, we defie B d B +, where B B + B i. Let B = B, B,, B, B,+,+ B B + γ,+ d B = B,, B, B,+, B,+ B B + γ,+. Now B B B i B out with equlity eig possile. Let F out = B out \ (B B ) d F i = B i \ (B B ). Without loss of geerlity, osider some D F i. The ridge D ot hve tthmets o the iterior of either γ,+ or γ,+ ; furthermore, D must hve tthmets o oth α = γ, γ, γ, d β = γ,+ γ +,+ γ +,+ d these must e ll of the tthmets of D. Note ow tht either the oly tthmet of D o α is v or the oly tthmet of D o β is v + (ssume the former) euse otherwise D would ross γ,+ i σ. So ow ll suh ridges i F i hve v s their sole tthmet o α, otherwise they would ross D i either σ or σ (depedig o whih side of the hord γ,+ they re o). Similrly, ll ridges i F out must hve either v s their sole tthmet o α or v + s their sole tthmet o β. Let α e the supth of α etwee the etreme edpoits of B. Let β e the supth of β etwee the etreme edpoits of B. Let H = H ( B ). Ay H-ridge eterior to C H i σ with tthmets produig H-pth tht would ross some H-pth of B if it were drw eterior to C H must e reemedded ridge. Let A e the olletio of suh H-ridges. We ow perform Q-twist higed o the edpoits of α d β i σ H d lthed t some verte γ,+. If this Q-Twist e eteded to ll of G i σ, the we oti emeddig σ for whih σ H = ν 0 d σ H = ν d the we go from σ to σ y sequee of Q-Twists s i Cse. If we ot perform this Q-Twist o ll of G i σ, the A. I Figure 8, we hve emples of where B A (the left ofigurtio) d where B A = (the right two ofigurtios). Let  = A B whe B A d let  = A whe B A =.

13 + Iside C H OR + Figure 8. ( ) Let H = H  d α d β eted α d β to the etreme tthmets of  o α d β. Ay H-ridge iterior to C H i σ with tthmets produig H-pth tht would ross some H-pth of (B Â) if it were drw iterior to C H must e reemedded ridge. Let A e the olletio of suh ridges. I Figure, the lk ridges i the first two ofigurtios form A d i the third ofigurtio we must hve tht A =. Iside C H OR + Iside C H Figure. + Iside C H OR + Iside C H OR + Iside C H We ow perform Q-twist higed o the edpoits of α d β i σ H d lthed t oe or two verties d. If this Q-Twist e eteded to ll of G i σ, the we oti emeddig σ for whih σ H = ν 0 or ν (depedig o whether or ot B is otied i  or ot) d σ H = ν d the we go from σ to σ y sequee of Q-Twists s i Cse or Cse. If we ot perform this Q-Twist o ll of G i σ, the A. If so, the let H = H ( A ) d α d β eted α d β to the etreme tthmets of A o α d β. Ay H-ridge eterior to C H i σ with tthmets produig H-pth tht would ross some H-pth of (B  A ) if it were drw eterior to C H must e reemedded ridge. Let A e the olletio of suh H-ridges. (I Figure, the first ofigurtio must hve A = ut the seod ofigurtio my hve A oempty.) If A =, we fiish y etedig to ll of G Q-Twist o σ H higed o the edpoits of α d β d lthed t d d the refer to either Cse or Cse. If A, the we defie  similrly to  d iterte this proess gi. This proess must ed s G is fiite. Cse : Suppose σ H = ν d σ H = ν j for j. See Figure 0. I Cse. sy tht d i Cse. sy tht =. Figure 0. Emeddigs of H i Cse +j + j + +j + + j Cse. Sie, there re oly two types of reemeddle ridges. Followig similr ottio s we did i Cse, the first type of reemeddle ridges prtitio ito the followig sets: B,, B,, B d B +,, B +,+, B + d B j,j, B j,j+, B j, d B +j,+j, B +j,+j+, B +j. Sy tht

14 ridges i the first si sets go log with γ,+ d ridges from the seod si sets go log with γ j,+j. The seod type of reemeddle ridges prtitio ito fs with pies from {v, v, v, v +, v +, v } or {v j, v j, v j+, v +j, v +j, v +j+ }. Sy tht ridges i fs with pe from the first set go log with γ,+ d ridges i fs with pe from the seod set go log with γ j,+j. So ow i σ ll reemeddle ridges goig log with γ j,+j must e eterior to C H eept those i B j B +j whih re must e iterior to C H. Similrly, i σ ll reemeddle ridges goig log with γ,+ must e eterior to C H eept those i B B + whih must e iterior to C H. Therefore there is emeddig σ of G with ll ridges goig log with γ j,+j d ll ridges goig log with γ,+ eterior to C H eept the ridges i B B + B j B +j whih re ll iterior to C H. Note tht σ H = ν 0 d so we go from σ to σ y sequee of Q-Twists s i Cse d we go from σ to σ y sequee of Q-Twists s i Cse. Cse. Here = d so σ H = ν 0 d d σ H = ν. See Figure for differet rederig of these emeddigs. Rell tht C H is the otgo o v v v v v v v v 8 whih is o the lower right i these figures. 8 8 Figure. I this se, fied H-ridge log with the fied sugrph H eed ot tully e fied poitwise i σ d σ. This is euse there my e H-ridges with ll their tthmets o γ, γ,8. So we do ot use the terms fied d reemedded i this se. We ow prtitio the types of H-ridges tht eist i oth emeddigs of H ito four differet lsses. First, sigulr ridge is sigle v v -, v v -, v v 8 -, or v v 8 -edge. For eh of these sigulr ridges we sy it hs orer verte v, v, v, or v, respetively. Seod, orer ridge hs ll tthmets o two djet rhes of H d is ot sigulr ridge. The rh verte iidet to oth rhes of orer ridge is lled the orer verte of the ridge. Third, tipodl ridge is ridge tht hs ll tthmets o C H tht is ot orer ridge or sigulr ridge. From Figure we see tht y tipodl ridge hs ll of its tthmets o some γ,+ γ +,+. Note tht, y the mimlity of V 8 mog ll V -sudivisios i G, suh ridge ot hve tthmets i the iteriors of oth γ,+ d γ +,+ d tht tipodl ridge is eterior to C H i oth emeddigs. Fourth, Peterse ridge is ot orer ridge or sigulr ridge d either hs tthmet i the iterior of oe of γ, d γ,8 or hs ll of its tthmets o t lest three verties from {v, v, v, v 8 }. Note tht Peterse ridge is lwys eterior to C H i oth emeddigs with ll tthmets o γ, γ,8. Further ote tht it is impossile to hve oth tipodl ridge d Peterse ridge, s oth re eterior to C H i oth emeddigs. We split the remider of this se ito five suses. I Cse.. we sy tht there is Peterse ridge with tthmets i the iteriors of oth γ, d γ,8. If there is o Peterse ridge hvig tthmets i the iteriors of oth γ, d γ,8, the without loss of geerlity we sy tht i Cse.. tht there is Peterse ridge with tthmets v, v 8, d v where v is iterior verte of γ,, i Cse.. tht there is Peterse ridge tthed t v 8 d v where v is iterior verte of γ,, i Cse.. tht there is Peterse ridge hvig three or four tthmets ll of whih re from {v, v, v, v 8 }, d i Cse.. tht there is o Peterse ridge. I Cses.... let B deote the Peterse ridge idetified. Cse.. There is H-pth i B whose uio with H forms sudivisio of the Peterse grph, ll this sudivisio P. Rehoose P so tht it hs o lol ridges d the sme rh verties. There

15 re two distit leled emeddigs of P i the projetive ple ( ft oe hek or see [8]) d we lim tht σ P σ P. If we ssume tht σ P = σ P, the sie y emeddig of P hs represettivity, y P -ridge elogs to uique fe of P d so y -oetivity the emeddig of P -ridge i its fe is uique up to isotopy d so σ = σ, otrditio. Now, oe hek tht y P -ridge i G must hve ll of its tthmets o two djet rhes of P i order to elog to fe of oth emeddigs of P. Agi we ow hve uique orer verte of P for y P -ridge. Oe lso hek tht for y uv-rh of P, tht the orer ridges for u d the orer ridges for v do ot overlp log the uv-rh or else they will ross i either σ or σ. Thus we go from σ to σ y P-Twist or degeerte P-Twist. Cse.. Here H B gi otis Peterse grph sudivisio d we fiish s i Cse... Cse.. Here H B otis sudivisio of the -edge otrtio of the Peterse grph, ll it P, where σ P d σ P re s show i Figure. Note tht y P -ridge is either H-ridge or hs tthmets o the iterior of the v 8 v -rh of P. Let H d P e the olletios of these two types of P -ridges. The v 8 v -rh my ow e rehose s i [,..] so tht there re o lol P -ridges o the v 8 v -rh d so sie H hs o lol ridges, either does P. 8 8 Figure. Rell tht there e o tipodl ridges i H d so y ridge i H is either orer ridge or sigulr ridge. Furthermore y orer ridge t v 8 must hve ll of its tthmets o γ,8 γ,8. Ay ridge i P is either orer ridge with orer verte v or hs ll tthmets o γ,8 γ 8,. Thus there is -edge deotrtio G of G t v 8 whih eteds the emeddigs σ d σ to σ d σ logous to wht is show i Figure. Note tht the figure depits oly orer ridges t v 8 d ot sigulr ridges. However, y sigulr ridge tht my eist e ssiged to either v 8 or v 8 s eeded. So ow G otis sudivisio of the Peterse grph d is -oeted d so σ d σ re relted y sigle P-Twist s i Cse... Thus σ d σ re relted y this sme P-Twist or P-Twist otied y otrtio of it. 8' 8 8' 8 Figure. Cse.. Here H B otis sudivisio of the -edge otrtio of the Peterse grph d so we fiish s i Cse... Cse.. If there re o sigulr H-ridges, the sie there re o Peterse ridges, the y H- ridge tht is iterior to C H i oth σ to σ or eterior to C H i oth σ to σ is fied poitwise (up to isotopy) with respet to H d so we fiish s i Cse.. So ssume there re sigulr ridges. Either there is sigulr ridge tht is eterior to C H i oth emeddigs or ot. If ot, the, similrly, y H-ridge tht is iterior to C H i oth σ to σ or eterior to C H i oth σ to σ is fied poitwise (up to isotopy) with respet to H d so we fiish s i Cse.. If so, the ssume, without loss of

16 geerlity, tht the sigulr ridge is v v 8 -edge. Let H e the uio of H d the v v 8 -edge. So σ H d σ H re s show i Figure. Figure. 8 Now i this se, we perform -edge deotrtio of G t v 8 similr to Cse.. tht eteds the emeddigs. Agi we get tht σ d σ re relted y sigle P-Twist. Proof of Lemm. for V 8 -free grphs I this setio, we will prove Lemm. i the se where G is V 8 -free. Formlly, we ssume tht G is -oeted d V 8 -free d hs two emeddigs σ d σ i the projetive ple. We will show tht these two emeddigs re relted y Q-Twists d P-Twists. We use Theorem. whih hrterizes iterlly -oeted grphs with o V 8 -mior. Hee we eed to lyze the -seprtios of G to redue to iterlly -oeted su-struture i G. Two degeerte ses re where G is -sum of two plr grphs (Setio.) d G is -sum of two o-plr grphs (Setio.). The mi lysis omes i Setio... A -sum of two plr grphs Let G e -oeted, o-plr grph tht is emedded i the projetive ple d G ot iterlly -oeted. The there is -sum G = G G. I this setio we will prove Lemm. for the se for whih G d G re oth plr. The proof follows from Propositios. d.. (We tully do ot use the ssumptio tht G is V 8 -free i this setio.) Rell tht yle i grph is peripherl if it is hordless d o-seprtig. Also rell tht i -oeted plr grph the peripherl yles re etly the fil yles. Propositio.. Suppose G is -oeted, projetive plr, oplr, d G = G G where oth G d G re plr. The the trigle of summtio is o-peripherl i t lest oe of G d G. Furthermore, G = K H H H k where k, eh H i is plr d summed ito K, d the trigle of summtio is peripherl i H i. Proof. If the trigle of summtio i G G is peripherl i oth, the G = G G is plr, otrditio. Supposig the trigle is ot peripherl i G, we get tht G = K G, G, G where the sums re ll t K. Repetig this proess we get tht G = K H H H k i whih eh H i is plr with its trigle of summtio eig peripherl d k. We ot hve tht k euse the G will oti K, -mior, whih is ot projetive plr, otrditio. Cosider the si emeddigs show o the left i Figure. Igorig the shded trigles, oe hek tht these si emeddigs re ll of the emeddigs of K, i the projetive ple. Eh rrow represets Q-Twist etwee the two emeddigs t its eds. For emple, to hge the emeddig t the top of the figure to the djet oe lokwise to it, use Q-twist higed t,,, d lthed t. Similrly o the right of Figure we show ll of the emeddigs d Q-Twists etwee them for K,. 8

17 Propositio.. If G = K H H H k is s i Propositio. d σ d σ re two emeddigs of G i the projetive ple, the we go from σ to σ y sequee of Q-Twists. Proof. Give G = K H H H k s i Propositio.. If k =, the G otis sudivisio H of K, with H-ridges s show o the left i Figure d so G hs the si emeddigs i the projetive ple s show. The rrows ow represet Q-Twists tht re modified from the oes for K,. For emple, to hge the emeddig t the top to the oe the djet oe lokwise to it, use Q-twist higed t,, d two ut-verties o the (, )-pth d (, )-pth d lthed t. For k =, G otis sudivisio H of K, d we get the struture d emeddigs of G s show o the right of Figure. These re relted y the sme Q-twists s i the se where k =. Figure.. A -sum of two o-plr grphs I this setio we prove Lemm. for the se tht G is -sum of two oplr grphs. The proof follows from Propositio.. (We tully do ot use the ssumptio tht G is V 8 -free i this setio.) Propositio.. If G is -oeted d G = G G where eh G i is oplr, the we go from y oe emeddig of G i the projetive ple to y other emeddig y sequee of Q-Twists. Proof. Sie eh G i is oplr d -oeted, either G i = K or G i otis K, -sudivisio. From [, 0..], if G i otis K, -sudivisio, the G i otis mior from Figure where the trigle of summtio of G i is show i old. Cll the left-hd grph T d the right-hd grph T. Figure. Neither G or G oti T -mior euse the if the other term is isomorphi to K or otis either T - or T -mior it will imply tht G = G G otis K, -mior. This mkes G ot projetive plr, otrditio. Thus for eh i, G i = K or G i otis T -mior. If G = G = K, the G = G G osists of K, log with two dditiol edges tht oet pirs of -vlet verties, d possily some edges of trigle o the -vlet verties. Let K, e the grph otied from K, y ddig (, )- d (, )-edges. Sie K, hs the si emeddigs s desried efore, the emeddigs of K, re otied from these si emeddigs. O the left of Figure, we show these si emeddigs of K,. The rrows etwee the emeddigs orrespod to Q-Twists similr to those i Propositio.. I the middle of Figure, we show the four emeddigs of K,

18 tht llow the (, )-edge. This (, )-edge hs two possile plemets show y dshed lies i the figure. Agi, the fleiilities re otied y similr Q-Twists d flippig of the (, )-edge. Filly, o the right of Figure, we show the two emeddigs of K, tht llow the (, )-edge d (, )-edge. These two edges eh hve two possile plemets show y dshed lies i the figure. Agi, these re relted y Q-Twists d flippig of the (, )- d (, )-edges. It is ot possile to emed ll three edges of the trigle {,, }. Figure. So ow we my ssume without loss of geerlity tht G otis T -mior. Lel the verties of K, s v, v, v, v, v, v, v s i Figure. Now if G = K, the G otis mior isomorphi to the grph otied from K, y sigle split t verte v levig -seprtio with v d v o oe side d v d v o the other side. Cll this grph S. We get the similr result whe G hs T -mior d the two miors i G d G re summed with the -vlet verties o the trigle of summtio idetified. If G d G oth hve T -miors d the sum does ot hve the -vlet verties oiidig, the G otis mior isomorphi to the grph otied from K, y splits t v d t v tht leve -seprtio with v d v o oe side d v d v o the other side. Cll this grph S. Note tht if we split the third -vlet verte of K, i wy tht preserves the seprtio of v, v from v, v, the we oti grph tht is eluded mior for projetive plrity. So we split the remider of the proof ito two ses: i Cse, G otis S -sudivisio s desried d i Cse, G otis S -sudivisio s desried. I eh se we ssume tht v, v G d v, v G. Cse Of the si emeddigs of K, (see Figure ) there re four tht eted to emeddigs of S. The ottom d lower left emeddigs i Figure do ot eted. By symmetry we eed oly show how to go from the emeddig of G with the emeddig of S orrespodig to the top emeddig i Figure to emeddig of G with the emeddig of S give y oe of the other three. For eh of the three ses, Figure 8 shows ll of the possile S -ridges tht our i oth emeddigs of S with -seprtio of G t v, v, v. Bridges roud the split verte v re show i lighter olors. The reder hek tht o other S -ridges re possile. I the first se, we go from the first emeddig to the seod y degeerte Q-Twist higed t v, v d lthed t v. I the seod se, we go from the first to the seod emeddig y Q-twist higed t v, v, v, v d lthed t v, v. I the third se, we go from the first emeddig to the seod emeddig y degeerte P-Twist otied from the etrl view of the P-Twist (see Figure ) with v i the eter pth, the pthes 0 d otrted to mke v, d pthes d otrted to mke v. 8

19 Figure 8. ' '' ' '' ' '' ' ' ' ' ' '' ' ' ' ' ' ' '' ' ' ' ' ' ' '' ' ' Cse Of the si emeddigs of K, (see Figure ) there re two tht eted to emeddigs of S. They re the two emeddigs o the right i Figure. The possile S -ridges tht our i oth emeddigs of S ll fll ito the shded regios of Figure. (Rell tht edge from the split t verte v d the edge from the split t verte v re o seprte sides of the -seprtio.) Figure. ' ' ' ' So ow we go from the first emeddig to the seod y degeerte Q-Twist higed t v d lthed t v. d. Redutio to iterlly -oeted frme I this setio we prove Lemm. for the se tht G is V 8 -free d ot e writte s -sum of two plr or two oplr terms. A -oeted oplr grph G dmits pth deompositio with iterlly -oeted frme F G d pthes P i whe either G is iterlly -oeted (d is its ow frme with o pthes) or G = F G P P P k where. F G is iterlly -oeted o-plr grph, eh P i is plr d summed ito trigle of F G, the trigle of summtio is peripherl i P i, o three P i re summed ito the sme trigle of F G. Propositio.. If G is -oeted, oplr, d G ot e writte s -sum of two plr or two oplr terms, the G dmits pth deompositio or y two emeddigs of G re relted y sequee of Q-Twists. Proof. We proeed y idutio o V (G) + E(G). I the se se V (G) + E(G) = d G = K or K, d our result is immedite. So ow sy tht V (G) + E(G) >. If G is iterlly -oeted, the we hve our result. If ot, the write G = G P where the summtio is log trigle T d E(P ). By ssumptio G is oplr d P is plr. Rehoose G d P so tht the umer of verties i P is miml. If T is ot peripherl i P, the euse P is plr we get tht P = P P where the -sum is o T d T is peripherl i eh P i.

20 Either G ot e writte s -sum of two plr or two oplr terms or it. Let these e Cse d Cse. Cse By the idutive hypothesis, we the get tht G = F G Q Q Q m s stted i our desired result. Let T i e the trigle i F G log whih the summtio with Q i is tke. By the mimlity of P d the ft tht G is ot -sum of two oplr grphs, it must e tht T is trigle of F G d is ot o the sme verties of y T i. Thus either G = F G Q Q Q m P or G = F G Q Q Q m P P stisfies our desired olusio. Cse We ssume tht G = G G where G d G re oth plr or oth oplr. (Assume tht T G.) It ot e the ltter se euse the G = G (G P ) where G d G P re oth oplr, otrditio. If oth G d G re plr, the rehoose G d G so tht there is mimum umer of verties i G d T G. Let T e the trigle of summtio for G G. Note tht sie G is ot -sum of two plr terms, it must e tht G P is oplr. Hee T is operipherl i either G or P. If T is operipherl i G d peripherl i P, the G = G G summed log T d where T is i G. Thus G = (G G ) (G P ) where (G P ) is plr. This otrdits the mimlity of P. So it must e tht T is ot peripherl i P d G = G G (P P ) (see Figure 0). By similr rgumet the mimlity of G implies tht T is operipherl i G d hee G = H H log T whih mkes G = (H H ) G (P P ). I Cses.,.,., d. sy tht V (T ) V (T ) = 0,,, d, respetively. I ll three ses, let V (T ) = {,, } d V (T ) = {,, }. T' T G H P G H P P Figure 0. Cse. The grph M ostruted y tkig two verte-disjoit opies of K, d oetig the - prtite sets y three edges is oe of the eluded miors for projetive-plr grphs. Sie G must e -oeted, there re three verte-disjoit pths likig V (T ) to V (T ). Thus G hs M-mior, otrditio. Cse. Sy tht V (T ) V (T ) = {}. Sie G is -oeted, there re disjoit pths i G \ likig {, } d {, } (ssume without loss of geerlity tht these two pths lik to d to ) d so G otis s S -sudivisio, ll it S. The two emeddigs of S re show i Figure. So i this se, the S-ridges i G fll ito the shded regios show i Figure where the S-ridges i G re show i drker grey. Ay S-ridge i H H P P is fied poitwise up to isotopy with respet to fied emeddig of S ; however, the ridges i G eh my hve up to two possile plemets i two of the drk shded regios. Thus the ofigurtio of the ridges i G d their possile reemeddigs re s desried i Cse... of Setio where the reemeddigs re show to e relted to eh other y sequees of Q-Twists. The two reemeddigs of the S -sudivisio re relted to eh other y sigle Q-Twist higed o, d lthed t. Figure. ' ' ' ' 0

21 Cse. Sy tht V (T ) V (T ) = {, }. I this se G otis S -sudivisio, ll it S, rooted o {,,, } d with -vlet verties v P, v P, v H, v H. Of the si emeddigs of K, show i Figure with orrespodig lels, oly four llow the deotrtio to S. The ottom d ottom left emeddigs of K, do ot eted to S. Now if v is verte i G \ {,,, }, the y -oetivity there re three iterlly-disjoit pths i G oetig v to {,,, }. I the four possile emeddigs of S, the oly possiilities re tht v is liked to {,, } or {,, }. Furthermore, if v liks to {,, }, the o other verte i G \ {,,, } liks to {,, }. If there re verties i G \ {,,, } d they ll lik to {,, }, the there re two possile emeddigs of S with ll of the S-ridges fllig ito the shded regios show o the left of Figure. Similr to Cse. ove, the S-ridges i H H P P re fied with respet to fied emeddig of S ut the ridges i G eh hve up to two possile plemets. So s i the previous se, y two emeddigs of G re relted y sequee of Q-Twists. If there re verties i G \ {,,, } d they ll lik to {,, }, the the emeddigs of S re s show o the right i Figure with S-ridges i the shded regios. Similr to wht is eplied i the previous prgrph, y two emeddigs of G re relted y Q-Twist ' Figure. ' ' ' If there re o verties i G \ {,,, }, the the S-ridges of G re ll orer ridges orered t v, v, v, v whih re ll fied poitwise with respet to give emeddig of the S -sudivisio sve for sigle-edge ridges o {,,, }. So sie there re four emeddigs of S i the projetive ple, ll emeddigs of G re show i Figure up to fleiility of sigle edges (whih is outed for y degeerte Q-Twists). The first emeddig goes to the seod y Q-Twist higed t, d lthed t. The first emeddig goes to the third y Q-Twist higed t, d lthed t, (the ridges i the twisted pth is show i drker grey). The third emeddig goes to the fourth y Q-Twist higed t, d lthed t. Figure. ' ' ' ' Cse. Here we must hve tht V (G ) = euse otherwise G will oti K, -mior, otrditio of projetive plrity. However ow if V (G ) =, the G = G P where oth G d P re plr, otrditio.

22 Now osider -oeted oplr grph G tht hs pth deompositio F G P P P k. Note tht F G is mior of G y -oetivity d the defiitio of -summig. (We do ot osider Y -Delt opertio to e -sum.) If G is emedded i the projetive ple, the the pth deompositio turlly yields uique emeddig (up to flippig sigle edges i F G tht re ot i G) of F G with trigulr shded pthes i the ples of the P i s. Similrly suh pth emeddig of F G orrespods to uique emeddig of G. So ow let ψ d ψ e two emeddigs of G d σ d σ ssoited pth emeddigs with fleile sigle edges pled ritrrily. Thus if we epli how to go from σ to σ y Q-Twists d P-Twists, the we will hve eplied how to go from ψ to ψ y Q-Twists d P-Twists. Thus Theorem. suffies to omplete the proof of Lemm. for the se where G is V 8 -free. Theorem.. If G is -oeted, oplr, V 8 -free, d hs pth deompositio, the we go from σ to σ y sequee of Q-Twists d P-Twists. The proof of Theorem. egis s follows. Sie G does ot oti V 8 -mior, either does F G (euse F G is mior of G) d so Theorem. yields ses for the et struture of F G. I Setio.., we ssume tht F G hs five verties; i Setio.., F G is doule wheel; i Setio.., F G hs si verties; i Setio.., F G hs seve verties; i Setio..8, F G is -verte overle; d i Setio.., F G = L(K, )... Frmes o five verties Give tht F G hs five verties d is oplr, we get tht F G = K. There re distit leled emeddigs of K o the projetive ple. Twelve of the hve fil -yle d the remiig do ot (see Figure ). Figure. Deote the emeddig with fil -yle,,, d, e y σ (de). We eed ot osider the emeddigs without fil -yle y the followig rgumet. If σ is s i the right of Figure, the euse edges (, ) d (, ) form the digols of qudrilterl fe (,,, ), t lest oe of (, ) or (, ) e flipped ross the oudry y degeerte Q-Twist euse there e t most two trigulr pthes i the fe (,,, ). Thus we oti ew pth emeddig σ with K hvig fil -yle. So for the remider of this setio, we ssume tht oth σ d σ re fil -yle type emeddigs. Of the emeddigs of K with fil -yle, the pthes fll ito two types: outside pthes hve their tthmets o oe of the trigulr fes of K d iside pthes tht do ot. Outside pthes my e emedded i either the iterior or eterior of the petgo, while iside pthes must e emedded i the iterior. Therefore, if doule pth ours, the they re outside pthes. I Cse sy there is doule pth d i Cse there is ot. Note tht there ot e two sets of doule pthes. Cse Assume the doule pth is o the verties (,, ). These two trigulr pthes re emedded so tht they meet t edge d t the tipodl verte from tht edge o the -fe. Moreover, either they meet i the sme edge of trigle (,, ) i oth σ d σ or ot. Without loss of geerlity i