Thickness and Colorability of Geometric Graphs

Transcription

1 Thicknss an Colorability o Gomtric Graphs Stphan Durochr 1 Dpartmnt o Computr Scinc, Univrsity o Manitoba, Winnipg, Canaa Elln Gthnr Dpartmnt o Computr Scinc, Univrsity o Colorao Dnvr, Colorao, USA Dbajyoti Monal 2, Dpartmnt o Computr Scinc, Univrsity o Manitoba, Winnipg, Canaa Abstract Th gomtric thicknss o a graph G is th smallst intgr t such that thr xist a straight-lin rawing Γ o G an a partition o its straight-lin gs into t substs, whr ach subst inucs a planar rawing in Γ. Ovr a ca ago, Hutchinson, Shrmr, an Vinc prov that any n-vrtx graph with gomtric thicknss two can hav at most 6n 18 gs, an or vry n 8 thy construct a gomtric thicknss-two graph with 6n 20 gs. In this papr, w construct gomtric thicknss-two graphs with 6n 19 gs or vry n 9, which improvs th prviously known 6n 20 lowr boun. W thn construct a thicknss-two graph with 10 vrtics that has gomtric thicknss thr, an prov that th problm o rcognizing gomtric thicknss-two graphs is NPhar, answring two qustions pos by Dillncourt, Eppstin an Hirschbrg. Finally, w prov th NP-harnss o coloring graphs o gomtric thicknss t with 4t 1 colors, which strngthns a rsult o McGra an Zito, whn t = 2. Kywors: Complt Graph, Gomtric Thicknss, Coloring. 1. Introuction Th thicknss θ(g) o a graph G is th smallst intgr t such that th gs o G can b partition into t substs, whr ach subst inucs a planar A prliminary vrsion o th papr appar in th Procings o th 39th Intrnational Workshop on Graph-Thortic Concpts in Computr Scinc (WG 2013) [12]. Corrsponing author arsss: urochr@cs.umanitoba.ca (Stphan Durochr), lln.gthnr@ucnvr.u (Elln Gthnr), jyoti@cs.umanitoba.ca (Dbajyoti Monal) 1 Work o th author is support in part by th Natural Scincs an Enginring Rsarch Council o Canaa (NSERC). 2 Work o th author is support in part by a Univrsity o Manitoba Grauat Fllowship. Prprint submitt to Computational Gomtry January 17, 2016

2 graph. Sinc 1963, whn Tutt [23] irst ormally introuc th notion o graph thicknss, this proprty o graphs has bn xtnsivly stui or its intrst rom both th thortical [2, 5, 7] an practical point o viw [19, 21]. A wi rang o applications in circuit layout sign an ntwork visualization, hav motivat th xamination o thicknss in th gomtric stting [7, 13, 16]. Th gomtric thicknss θ(g) o a graph G is th smallst intgr t such that thr xist a straight-lin rawing (i.., a rawing on th Euclian plan, whr vry vrtx is rawn as a point an vry g is rawn as a straight lin sgmnt) Γ o G an a partition o its straight-lin gs into t substs, whr ach subst inucs a planar rawing in Γ. I t = 2, thn G is call a gomtric thicknsstwo graph (or, a oubly-linar graph [16]), an Γ is call a gomtric thicknsstwo rprsntation o G. Whil thicknss os not impos any rstriction on th placmnt o th vrtics in ach planar layr, gomtric thicknss orcs th sam vrtics in irnt planar layrs to shar a ix point in th plan. Eppstin [13] clarly stablish this irnc by constructing thicknss-thr graphs that hav arbitrarily larg gomtric thicknss Structural Proprtis Gomtric thicknss has bn broaly xamin on svral classs o graphs,.g., complt graphs [7], boun-gr graphs [4, 11, 13], an graphs with boun trwith [8, 10]. Hutchinson, Shrmr, an Vinc [16] xamin proprtis o graphs with gomtric thicknss two. Thy prov that ths graphs can hav at most 6n 18 gs, an or vry n 8 thy construct a gomtric thicknss-two graph with 6n 20 gs. Th graphs that gav th 6n 20 lowr boun wr rctangl visibility graphs, i.., ths graphs can b rprsnt such that th vrtics ar axis-align rctangls on th plan with ajacncy trmin by th horizontal an vrtical visibility. Hutchinson t al. [16] prov that a rctangl visibility graph can hav at most 6n 20 gs, thror, any gomtric thicknss-two graph with mor than 6n 20 gs (i xists) cannot b a rctangl visibility graph. Evn atr svral attmpts [7, 11] to unrstan th structural proprtis o gomtric thicknss-two graphs, th qustion whthr thr xists a gomtric thicknss two graph with 6n 18 gs rmain opn or ovr a ca. Answring this qustion is quit challnging sinc although on can gnrat many thicknss-two graphs with 6n 18 or 6n 19 gs, no icint algorithm is known that can trmin th gomtric thicknss o such a graph. Howvr, by xamining th point conigurations that ar likly to support gomtric thicknss-two graphs with larg numbrs o gs, w hav bn abl to construct gomtric thicknss-two graphs with 6n 19 gs, which improvs th prviously known 6n 20 lowr boun on th maximum numbr o gs that a graph with gomtric thicknss two can hav. In Sction 2 w hav shown that th K 9 minus an g is a thicknss-two graph, which has 6n 19 gs. W thn show that thicknss-two graphs that o not contain K 9 minus an g may also hav larg numbr o gs. Thorm 1 For ach n 9, thr xists a gomtric thicknss-two graph with n vrtics an 6n 19 gs that contains K 9 minus an g as a subgraph. For 2

3 ach n 11, thr xists a gomtric thicknss-two graph with 6n 19 gs that os not contain K Rcognition Although thicknss is known or all complt graphs [2] an complt bipartit graphs [5], gomtric thicknss or ths graph classs is not compltly charactriz. Dillncourt, Eppstin an Hirschbrg [7] prov an n/4 uppr boun on th gomtric thicknss o K n, giving a nic construction or rprsntations with gomtric thicknss t = n/4. Thy also gav a lowr boun on th gomtric thicknss o complt graphs that matchs th uppr boun or svral smallr valus o n. Thir bouns show that th gomtric thicknss o K 15 is gratr than its thicknss, i.., θ(k 15 ) = 4 > θ(k 15 ) = 3, which sttls th conjctur o [18] on th rlation btwn thicknss an gomtric thicknss. Sinc th xact valus o θ(k 13 ) an θ(k 14 ) ar still unknown, Dillncourt t al. [7] hop that th rlation θ(g) > θ(g) coul b stablish with a graph o smallr carinality. In Sction 3 w prov that th smallst such graph contains 10 vrtics. Thorm 2 For vry n 9 an vry graph G on n vrtics, θ(g) = θ(g). For vry graph n > 10, thr xists a graph G on n vrtics such that θ(g) > θ(g). Sinc trmining th thicknss o an arbitrary graph is NP-har [19], Dillncourt t al. [7] suspct that trmining gomtric thicknss might b also NP-har, an mntion it as an opn problm. Th harnss proo o Mansil [19] rlis havily on th act that θ(k 6,8 ) = 2. Dillncourt t al. [7] mntion that this proo cannot b immiatly aapt to prov th harnss o th problm o rcognizing gomtric thicknss-two graphs by showing that θ(k 6,8 ) = 3. This complxity qustion has bn rpat svral tims in th litratur [8, 13] sinc 2000, an also appar as on o th slct opn qustions in th 11th Intrnational Symposium on Graph Drawing (GD 2003) [6]. In Sction 4 w sttl th qustion by proving th problm to b NP-har. Thorm 3 It is NP-har to trmin whthr th gomtric thicknss o an arbitrary graph is at most two Colorability As a natural gnralization o th wll-known Four Color Thorm or planar graphs [3], a long-staning opn problm in graph thory is to trmin th rlation btwn thicknss an colorability [17, 22]. For vry t 3, th bst known lowr boun on th chromatic numbr o th graphs with thicknss t is 6t 2, which can b achiv by th largst complt graph o thicknss t. On th othr han, vry graph with thicknss t is (6t)-colorabl [17]. Rcntly, McGra an Zito [20] xamin a variation o this problm that givn a planar graph an a partition o its vrtics into substs o at most r vrtics, asks to assign a color (rom a st o s colors) to ach subst such that two ajacnt vrtics in 3

4 irnt substs rciv irnt colors. Thy prov that th problm is NPcomplt whn r = 2 (rspctivly, r > 2) an s 6 (rspctivly, s 6r 4) colors. In Sction 5 w prov th NP-harnss o coloring gomtric thicknss-t graphs with 4t 1 colors. As a corollary, w strngthn th rsult o McGra an Zito [20] that coloring thicknss-(t = r = 2) graphs with 6 colors is NP-har. Our harnss rsult is particularly intrsting sinc no gomtric thicknss-t graph with chromatic numbr mor than 4t is known. Thorm 4 It is NP-har to color an arbitrary gomtric thicknss-t graph with 4t 1 colors. 2. Gomtric Thicknss-Two Graphs with 6n 19 Egs Lt K 9 b th graph obtain by lting an g rom K 9. In this sction w irst construct a gomtric thicknss-two rprsntation Γ o K 9, which has 6n 19 gs. W thn show how to a vrtics an gs in Γ such that or any n 9 on can construct a gomtric thicknss-two graph with 6n 19 gs. Although ths graphs contain K 9 as a subgraph, w prov that this is not a ncssary conition, i.., w also construct gomtric thicknss-two graphs with 6n 19 gs that o not contain K 9. Hutchinson t al. [16, Thorm 6] prov that i any gomtric thicknss-two graph with 6n 18 gs xists, thn th convx hull o its gomtric thicknss-two rprsntation must b a triangl. This rprsntation is quivalnt to th union o two plan triangulations that shar at last six common gs, i.., th thr outr gs an th othr thr gs ar ajacnt to th thr outr vrtics, as shown in black in Figur 1(a). Sinc ach triangulation contains 3n 6 gs, th uppr boun o 2(3n 6) 6 = 6n 18 ollows. Ths proprtis o an g maximal gomtric thicknss-two rprsntation motivat us to xamin pairs o triangulations that crat many g crossings whn rawn simultanously. In particular, w irst crat a st o points intrior to th convx hull such that aition o straight lin sgmnts rom ach intrior point to th thr points on th convx hull crats two plan rawings that, whil rawn simultanously, contain a crossing in all but th six common gs. Figur 1(b) illustrats such a scnario. W thn tri to xtn ach o ths two planar rawings to a triangulation by aing nw gs such that vry nw g crosss at last on initial g. W oun multipl istinct point sts or which all but on nwly a g cross at last on initial g, rsulting in multipl istinct gomtric thicknss-two rprsntations with 2(3n 6) 7 = 6n 19 gs. For xampl, s Figur 1(c), whr th unrlying graph is K 9, whr K 9 = K 9 \ (, ). W now introuc a w mor initions. Lt Γ b a gomtric thicknsstwo rprsntation. A triangl in Γ is mpty i it contains xactly thr vrtics on its bounary, but os not contain any vrtx in its propr intrior,.g., th triangl ghi in Figur 1(c). A quarangl Q in Γ crat by th intrsction o two mpty triangls is call r i nithr th intrior nor th bounary o Q contains any vrtx o Γ, as shown in Figur 1(c) in ark-grn sha. 4

5 a g (a) b a i h (b) c g i h b (c) c Figur 1: (a) Illustration or th shar gs (bol). (b) Initial point st. (c) A gomtric thicknss-two rprsntation Γ o K 9, whr K 9 = K 9 \ (, ). Th planar layrs ar shown in r (ash) an blu (thin). Black (bol) gs can blong to ithr r or blu layr. Fr quarangls ar shown in ark-grn (sha). Som gs ar rawn with bns or clarity. Thorm 1. For ach n 9, thr xists a gomtric thicknss-two graph with n vrtics an 6n 19 gs that contains K 9 minus an g as a subgraph. For ach n 11, thr xists a gomtric thicknss-two graph with 6n 19 gs that os not contain K 8. Proo. Sinc K 9 has a gomtric thicknss-two rprsntation, as shown in Figur 1(c), th claim hols or n = 9. W now claim that givn an n-vrtx gomtric thicknss-two rprsntation with 6n 19 gs that contains a r quarangl, on can construct a gomtric thicknss-two rprsntation with n + 1 vrtics an 6(n + 1) 19 gs. On can vriy this claim as ollows. Plac a nw vrtx p intrior to th r quarangl. Sinc a r quarangl is th intrsction o two mpty triangls, on can a thr straight lin gs rom p to th thr vrtics o ach mpty triangl such that th nw rawing 5

6 a k j h g i (a) b (c) a c (vrtx, color) h k j g i (a, 1) (g, 6) (b, 7) (h, 5) (c, 2) (i, 5) (, 3) (j, 7) (, 4) (k, 6) (, 4) (b) b () c Figur 2: (a) (b) Aing vrtics to a gomtric thicknss-two rawing. (c) () A graph with 11 vrtics, 47 gs an gomtric thicknss two that os not contain K 8. 6

7 in ach layr rmains planar, as shown in Figur 2(a). Sinc th numbr o vrtics incrass by on, an th numbr o gs incrass by six, th rsulting gomtric thicknss-two rprsntation must hav 6n = 6(n + 1) 19 gs. Obsrv that thr ar at last thr r quarangls in th gomtric thicknss-two rprsntation o K 9, as shown in Figur 1(c). Furthrmor, ths quarangls ar inpnnt o ach othr, i.., th mpty triangls corrsponing to ach quarangl ar irnt than th mpty triangls o th othr two quarangls. Thror, or ach i, whr 9 i 12, w can construct a gomtric thicknss-two rprsntation Γ i with i vrtics an 6i 19 gs that contains at last on r quarangl. W us ths our gomtric thicknss-two rprsntations as th bas cas, an assum inuctivly that or ach 9 i < n thr xists a gomtric thicknss-two rprsntation Γ i with i vrtics an 6i 19 gs that contains at last on r quarangl. W now construct a gomtric thicknss-two rprsntation with n vrtics an 6n 19 gs that contains a r quarangl. By inuction, Γ n 4 has a r quarangl. W a our vrtics to this quarangl an complt th triangulation in ach planar layr by aing 24 nw gs in total, as shown in Figur 2(b). Consquntly, th nw gomtric thicknss-two rprsntation Γ n contains 6(n 4) = 6n 19 gs. Sinc th nwly a gs crat nw r quarangls, Γ n also contains a r quarangl. For ach n 11, th proo or th claim that thr xists a gomtric thicknss-two graph with 6n 19 gs that os not contain any K 8 is similar. Figur 2(c) illustrats a gomtric thicknss-two rprsntation with 11 vrtics an 47 gs that contains at last thr inpnnt r quarangls: {( cik, agj), ( chj, aig), ( hj, b i)}. Th unrlying graph os now contain K 8 as a subgraph sinc it is 7-colorabl. To construct a gomtric thicknss-two rprsntation Γ or a largr valu o n, w a a K 8 -r subgraph intrior to a r quarangl in th sam way as in th arlir part o th proo (s Figurs 2(a) (b)). Hnc th rsulting graph must also b a K 8 -r graph. 3. Thicknss-Two Graphs with θ(g) 3 Dillncourt t al. [7] show that θ(k 15 ) = 4 > θ(k 15 ) = 3 an θ(k 6,8 ) = 3 > θ(k 6,8 ) = 2, an ask to trmin th smallst graph G such that θ(g) > θ(g). In this sction w show that th smallst graph G such that θ(g) > θ(g) contains tn vrtics, an construct such a gomtric thicknss-thr graph. To stablish this rsult w numrat all possibl gomtric thicknss-two rawings o K 9 using Aichholzr t al. s [1] point-st orr-typ atabas 3. Figur 3 illustrats all th thr combinatorially irnt conigurations o nin points that support gomtric thicknss-two rawings o K 9. It might initially 3 Th co is availabl onlin: DrawK9MinusOnEg.java 7

8 appar that Figurs 3(a) an (b) ar th sam. Howvr, obsrv that g lis on th lt hal-plan o (, ) in Figur 3(a) an on th right-hal plan o (, ) in Figur 3(b). W numrat ths gomtric thicknss-two rprsntations by prorming th stps S 1 an S 2 blow or vry point-st orr-typ P that consists o nin points. W us th concpt o intrsction graphs o sgmnts: Givn a st o straight lin sgmnts L, th propr intrsction graph G o L consists o L vrtics, whr ach vrtx corrspons to a istinct lin sgmnt in L, an two vrtics o G ar ajacnt i an only i th corrsponing straight lin sgmnts proprly cross. S 1. Construct a straight-lin rawing Γ o K 9 on P. S 2. For vry g in Γ, xcut th ollowing. - Dlt an tst whthr th propr intrsction graph trmin by th rmaining straight lin sgmnts is 2-colorabl. I th graph is 2-colorabl, thn Γ is a gomtric thicknss-two rprsntation o K 9. - Rinsrt in Γ. Lt Γ i, 1 i 3, b th rawings o K 9 pict in Figurs 3(a) (c), rspctivly. Th svn black (bol) gs in ach o ths rawings o not contain any crossing, i.., ths gs ar shar in both triangulations. By E i an E i w not th st o all gs, an th st o black gs in Γ i, rspctivly. Lt E i = E i \ E i. W vriy that th partition o th gs o E i into r an blu is uniqu by chcking that th propr intrsction graph G i o E i is connct. Figur 4(a) shows a spanning tr o G 1 o Γ 1. Fact 1. Lt Γ b a gomtric thicknss-two rprsntation o K 9. Thn th partition o th straight-lin gs o Γ, xcpt th svn gs that o not contain any propr crossing, into two planar layrs is uniqu. Morovr, th unsaturat vrtics ar connct in ach layr o Γ. W now catgoriz th vrtics o a K 9 into two typs: unsaturat (vrtics o gr 7), an saturat (vrtics o gr 8). Th vrtics an o Figurs 3(a) (c) ar unsaturat, an all othr vrtics ar saturat vrtics. Tak a nw vrtx an mak it ajacnt to th two unsaturat vrtics an any iv saturat vrtics o a K 9. Lt G s not th rsulting graph with 10 vrtics. Th ollowing thorm shows that θ(g s ) = 3 > θ(g s ) = 2. Th ia o th proo is irst to show a thicknss-two rprsntation o G s, an thn to show that G s contains a vrtx v that is not straight-lin visibl to all o its nighbors in any gomtric thicknss-two rprsntation o G s \ v. Finally, th proo shows that or vry graph G with at most 9 vrtics, θ(g) = θ(g). Thorm 2. For vry n 9 an vry graph G on n vrtics, θ(g) = θ(g). For vry graph n > 10, thr xists a graph G on n vrtics such that θ(g) > θ(g). 8

9 a g g h b (a) i h c a g g h h b i c (b) 9

10 a i b g (c) h c Figur 3: (a) (c) Gomtric thicknss-two rprsntations o K 9, whr K 9 = K 9 \ (, ). Egs ar rawn with polylins or clarity. Proo. W irst prov that th thicknss o G s is two. Lt v b th vrtx o gr svn in G s such that G s \v is a K 9. Tak th gomtric thicknss-two rprsntation o K 9, as shown in Figur 3(a), an plac th vrtx v intrior to th intrsction o th triangls bci an bc. It is now straightorwar to a th gs (with curv lins) rom v to th two unsaturat vrtics an iv saturat vrtics o K 9 maintaining th planarity o ach layr. S Figur 4(b). W now prov that th gomtric thicknss o G s is thr. Sinc G s contains a K 9, any gomtric thicknss-two rprsntation Γ o G s must contain a gomtric thicknss-two rprsntation Γ o K 9 rom Figurs 3(a) (c). Sinc Γ contains Γ, v cannot li on any straight lin sgmnt o Γ. A vrtx u o a planar rawing is straight-lin visibl to a point p i th straight lin btwn 10

11 a i b b b g b h i i a h a h g h c g c a g a g i g g i c g, h c h i b c i (a) a g h b v (b) i c Figur 4: (a) A spanning tr o G 1. (b) Th thicknss o G s is two. p an u os not cross any g or vrtx o th rawing. W now prov that at most six vrtics o Γ can b straight-lin visibl to a common point, an hnc v cannot b ajacnt to svn vrtics. Dlt all th black gs, i.., th gs common to both triangulations, rom Γ. Lt Γ not th rsulting rawing. Figurs 5(a) an (b) show th caniat rawings that ar obtain rom Figurs 3(a) an (c), rspctivly. W o not xamin Figur 3(b) sparatly sinc its clos rgions ar similar to that o Figur 3(a). In ach planar layr o Γ, v must li on som boun rgion or on th unboun rgion. Obsrv that th boun rgions in ach planar layr ar a collction o triangls an quarangls. I v lis intrior to an mpty triangl in ach planar layr, thn it can hav at most six ajacnt gs. Thror, v must li on th unboun rgion or on a quarangl o som planar layr. Each such rgion Q is unill in Figur 5. Examining vry such cas rvals that no point in ths rgions can hav straight-lin visibility to svn istinct vrtics o Γ, an hnc θ(g s ) = 3. Th tails o th cas by cas analysis ar oun in Appnix A. Hr w illustrat an xampl in th contxt o Figur 5(a), whr th quarangl Q = ihg that inclus v lis in th blu layr. On can partition Q into 5 rgions Q 1, Q 2,..., Q 5, pning on how th nighboring sgmnts o h intrscts Q. Lt q b a point in Q. I q Q 1 Q 2, thn sinc {, g} an {i} li on th opposit sis o th lin trmin by sgmnt hb, vrtx g cannot b visibl to q. Thror, q can s only 3 points 11

12 a b Q 1 g Q 2 Q 5 h c i g h i b (a) c a i b g (b) h c Figur 5: (a) (b) Illustration or Γ. 12

13 in th blu layr. Obsrv now that q is ithr insi a triangl in th r layr, or insi th quarangl ch. I q is insi a triangl in th r layr, thn q can s at most six vrtics in total, an hnc assum that q lis insi ch. Sinc {, h} an {, c} li on th opposit sis o th lin trmin by sgmnt ig, vrtx h cannot b visibl to q. Thror, also in this cas q can s at most six vrtics in total. Th cas whn q Q 4 Q 5 can b analyz in a similar ashion, whr an g cannot b visibl to q simultanously, an similarly h an cannot b visibl to q simultanously. In th rmaining cas w hav q Q 3, whr q can s all th vrtics {i, g, h, } in th blu layr. Sinc q is on th outr ac in th r layr, only th vrtics {b, c, i, h} ar visibl to q. Thror, q has straight-lin visibility to at most six points in total. Finally, obsrv that or vry graph G with at most nin vrtics, θ(g) = θ(g), as ollows. Sinc θ(g) = 1 i an only i θ(g) = 1 [21], w assum that θ(g) = 2. Sinc θ(k 9) = 2, i G is a subgraph o K 9, thn θ(g) = θ(g) = 2. Othrwis, G = K 9, whr θ(k 9 ) = θ(g) = 3. Thror, vry graph G with θ(g) > θ(g) must hav at last 10 vrtics. 4. Gomtric Thicknss-Two Graph Rcognition Mansil [19] show that th problm o rcognizing thicknss-two graphs is NP-har. In this sction w prov that trmining whthr θ(g) 2 is NP-har. For any input graph G = (V, E), w construct anothr corrsponing graph G such that G is a graph o thicknss two i an only i th gomtric thicknss o G is two. W irst prsnt som prliminary rsults, which will b usul to scrib th construction o G. For th clarity o th prsntation, proos o som o th lmmas ar givn in Appnix B. Lt G 1, G 2,..., G k b k 9 copis o K 9. Lt 1, 1 b th unsaturat vrtics o G 1. For ach G j, j > 1, mak 1 ajacnt to som unsaturat vrtx j o G j. Rr to th rmaining unsaturat vrtx o G j as j. A a vrtx v an mak v ajacnt to all th unsaturat vrtics o G 1, G 2,..., G k. Lt H k not th rsulting graph, which w rr to as a rigi graph. Th ollowing lmma scribs som proprtis o H k, whos proo is inclu in Appnix B. Lmma 1. Lt H k b a rigi graph. Thn in any thicknss-two rawing Γ o H k, th subgraph G inuc by th gs (v, 1 ), (v, j ) an ( 1, j ), whr 1 < j k, lis in th sam layr. Obsrv that G is a bipartit graph K 2,k 1 with vrtx-partition {v, 1 }, { 2,..., k }, plus th g (v, 1 ). W call th graph G th cor graph an th vrtx v th pol vrtx. Lt H b a chain o thr rigi graphs H a, H b, H c, ach a copy o H 17. For any H q, whr q {a, b, c}, lt { q i, q i }, whr 1 i 17, b th unsaturat vrtics o H q, an lt v q b th pol o H q. W now a th gs ( a 2, b 2), ( a 3, b 3),..., ( a 9, b 9) an ( c 2, b 10), ( c 3, b 11),..., ( c 9, b 17). Lt H not th rsulting graph. Figurs 6(a) (b) illustrat a schmatic rprsntation o H. 13

14 Th ollowing lmma scribs som proprtis o H, whos proo is inclu in Appnix B. Lmma 2. In any thicknss-two rawing Γ o H, thr xists a path rom v a,..., v b,..., v c that lis on th sam layr. It is straightorwar to xtn th abov lmma or a chain o mor than thr rigi graphs, as stat blow. Corollary 1. Lt H b a chain o q 3 rigi graphs H 1, H 2,..., H q, ach a copy o H 17. Lt v i b th pol o H i, whr 1 i q. Thn in any thicknsstwo rawing Γ o H, thr xists a path rom v 1,..., v 2,..., v q that lis on th sam layr. W ar now ray to scrib th NP-harnss rsult. Thorm 3. It is NP-har to trmin whthr th gomtric thicknss o an arbitrary graph is at most two. Proo. W ruc th problm o trmining thicknss-two graphs, which is NP-complt [19], to rcognition o gomtric thicknss-two graphs. For any input graph G = (V, E), w construct anothr corrsponing graph G such that G is a graph o thicknss two i an only i th gomtric thicknss o G is two. W construct G by rplacing ach g (v, w) by a chain H vw o our rigi graphs H 1, H 2,..., H 4, ach a copy o H 17, such that th vrtics v an w coinci to pols o H 1 an H 4. Figurs 7(a) an (b) pict an input graph G an a schmatic rprsntation o th corrsponing graph G, rspctivly. It is straightorwar to construct G in polynomial tim. W now show that G is a graph o thicknss two i an only i G amits a gomtric thicknss-two rprsntation. First assum that th thicknss o G is two, an lt {E r, E b } b th corrsponing partition o th gs, i.., E = (E r E b ) an E r E b = φ. W now comput a gomtric thicknss-two rprsntation o G. Not that th graphs G r = (V, E r ) an G b = (V, E b ) ar planar. Thror, w can us th algorithm o Ertn an Kobourov [14, 15] to comput a rawing o G on an O( V 3 ) O( V 3 ) gri R such that th ollowing proprtis hol: P 1. No two gs o E r or E b cross. P 2. Each g is rawn as polygonal chain with at most two bns, an P 3. Th vrtics an bns li on som intgr gri point o R. Figur 7(c) illustrats an xampl. W now rplac ach g (v, w) o G with th corrsponing chain H vw such that th gs btwn any pair o rigi graphs li on th sam layr as (v, w). W raw th rigi graphs aroun th vrtics an bn points such that thy o not intrr with th othr gs o th rawing. Figur 7() picts a schmatic gomtric thicknss-two rprsntation o G which corrspons to th rawing o Figur 7(c). 14

15 a g towars anothr rigi graph towars pol vrtx b i h c towars pol vrtx v b (a) b 9 b 10 b 17 b 1 b 1 c 9 c 1 c 1 a 1 v c a 1 a 9 v a (b) Figur 6: (a) Connctions o K 9 insi an outsi o a rigi graph. (b) A schmatic rprsntation o H. 15

16 a b c a c (a) b b a (b) c (c) () Figur 7: (a) An input graph G. (b) A schmatic rprsntation o G. (c) A thicknss-two rawing o G. () Th corrsponing gomtric thicknss-two rprsntation o G. 16

17 Assum now that G amits a gomtric thicknss-two rprsntation Γ. W show how to comput a thicknss-two rawing o G. By Corollary 1, or ach g (v, w) in G, thr xists a path P = (v,..., w) in H vw such that all gs o P li on th sam layr o Γ. W lt all gs o H vw xcpt th gs o P, an any rsulting isolat vrtics. Sinc ltion o vrtics an gs rom a gomtric thicknss-two rawing os not incras gomtric thicknss, w n up with a thicknss-two rprsntation o G. Not that w o not yt know whthr th problm o ining gomtric thicknss is in NP. Givn a crtiicat,.g., a partition o th gs o th givn graph G into t sts, it is not clar how to trmin whthr th graphs inuc by ach g st amit a simultanous gomtric mbing (which woul imply that th gomtric thicknss o G is at most t). 5. NP-harnss o Colorability In this sction w show th NP-harnss o coloring a graph with gomtric thicknss t with 4t 1 colors. By I(G, T, C) w not th problm o coloring a graph G with C colors, whr θ(g) T. W irst introuc a w initions. A join btwn two graphs is an opration that givn two graphs, as all possibl gs that connct th vrtics o on graph with th vrtics o th othr graph. By G t w not a class o thicknss-t graphs that satisis th ollowing conitions: 1. G 1 is th class o planar graphs. 2. I t>1, thn G t consists o th graphs obtain by taking a join o K 2 an G, whr G G t 1. W now hav th ollowing lmma. Lmma 3. It is NP-har to color an arbitrary graph G G t with 2t + 1 colors. Proo. I t = 1, thn coloring a planar graph (i.., t = 1) with 2t+1 = 3 colors is NP-har [17]. Assum inuctivly that th claim hols or any thicknss lss than t. W now prov th harnss or thicknss t, whr t > 1, as ollows. Givn an instanc I(G, t 1, 2(t 1) + 1) whr G G t 1, w construct a graph H by joining K 2 with a copy o G. Obsrv that H G t. It is now straightorwar to obtain a gomtric thicknss-t rprsntation or H by aing th vrtics o K 2 to a thicknss-(t 1) rprsntation o G (on to th lt an th othr to th right o th rprsntation) such that th gs ajacnt to th vrtics o K 2 li in th t-th planar layr but o not crat any propr g crossings. Thror, θ(h) t, an it now suics to prov that G is (2(t 1) + 1)-colorabl i an only i H is (2t + 1)-colorabl. I G is (2(t 1) + 1)-colorabl, thn w can color th copy o G that is contain in H with 2(t 1) + 1 colors. Finally, w color th vrtics o K 2 with two nw colors, an thus obtain a 2t + 1 coloring or H. 17

18 On th othr han, i H is (2t + 1)-colorabl, thn th vrtics o K 2 must hav two irnt colors. Th vrtics o H that blong to th copy o G cannot us ths two colors. Thror, G must b (2t + 1 2)-colorabl, i.., (2(t 1) + 1)-colorabl. Not rom th proo o Lmma 3 that θ(g t ) t. W us Lmma 3 to prov th NP-harnss o coloring gomtric thicknss-t graphs with 4t 1 colors. W mploy inuction on t. I t = 1, thn coloring a planar graph (i.., t = 1) with 4t 1 = 3 colors is NP-har [17]. W now assum inuctivly that or any t < t, it is NP-har to color a gomtric thicknss-t graph with 4t 1 colors. To prov th harnss o coloring a gomtric thicknss-t graph with 4t 1 colors, w ruc th harnss o coloring a gomtric thicknss-(t 1) graph with 2(t 1) + 1 colors. Givn an instanc I(G, t 1, 2(t 1) + 1), whr G G t 1, w construct a graph H(G, t) such that θ(h(g, t)) t an H(G, t) is (4t 1)-colorabl i an only i G is (2(t 1) + 1)-colorabl Construction o H(G, t) Lt th numbr o vrtics in G b n. Tak n copis H 1, H 2,..., H n o K 2t, an join ach vrtx o G with a istinct H i, 1 i n. Finally, tak a copy H o K 2t 1 an join it with vry H i. Lt th rsulting graph b H(G, t). To prov that θ(h(g, t)) = t, w irst rviw a construction o Dillncourt t al. [7] that givs a thicknss-t rprsntation o K 4t. Dillncourt t al. [7] prov that th 4t vrtics o a K 4t can b arrang in two rings o 2t vrtics ach, an outr ring an an innr ring, such that it can b mb using xactly t planar layrs. Th vrtics o th innr ring ar arrang to orm a rgular 2t-gon. For ach pair o iamtrically opposit vrtics, a zigzag path is construct as illustrat in Figur 8(a). This path has xactly on iagonal conncting iamtrically opposit points (i.., th iagonal conncting th two gray points in th igur.) Th union o ths zigzag paths, takn ovr all t pairs o iamtrically opposit vrtics, contains all th gs o K 2t in th innr ring, as shown in Figur 8(b). Consir now any zigzag path Z. For ach pair o iamtrically opposit vrtics, w can raw rays in two opposit irctions, so that non o th rays crosss any g o Z. Ths rays, in ach irction, mt at a common point (.g., p or q) orming th outr ring, as shown in Figur 8(c). Lmma 4. θ(h(g, t)) t, whr t > 1 an G G t 1. Proo. W comput a gomtric thicknss-t rprsntation o H(G, t), as ollows. Sinc G G t 1, θ(g) t 1. Tak a gomtric thicknss-(t 1) rprsntation o G an rotat it (i ncssary) such that no two vrtics li on th sam vrtical lin. Lt Γ not th rsulting rawing. For th ith vrtx in Γ, consir a thin vrtical strip S i through it, as shown in Figur 8() in gray. Consir a horizontal strip L blow Γ that intrscts all th vrtical strips. For ach i, w construct an innr ring (shown in black isk) that lis insi th intrsction o S i an L, whr this ring corrspons to th rawing o H i. 18

19 v 1 v 2 (a) (b) Outr Ring p Γ v 3 Innr Ring H 1 H2 H 3 L q (c) () Figur 8: (a) (c) Dillncourt t al. s construction [7]. (a) A zigzag path in th innr ring. (b) K 2t, whr t=3. (c) K 4t, whr t=3. () Th gomtric thicknss-thr rprsntation o H(G, t), whr t=3. Each subgraph H i is trmin by an innr ring, shown in black ot insi th horizontal strip L. Th vrtics o th outr ring ar shown in unill circl. Th gs that connct th vrtics o G with th vrtics o H i (i.., th gs in th vrtical strips) li in th t-th layr. Not that th innr rings must b scal own small nough such that ths gs o not crat any g crossing in any planar layr. Now construct an outr ring as in Dillncourt t al. s construction [7], an lt a vrtx rom th ring to obtain a gomtric thicknss-t rawing o H, as shown in Figur 8() Ruction Givn a gomtric thicknss-t graph G an a crtiicat coloring o G, on can chck in polynomial tim whthr th numbr o colors us is at most 4t 1, an whthr ach g o G rcivs two irnt colors at its n vrtics. Thror, th problm o coloring gomtric thicknss-t graphs with 19

20 4t 1 colors is in NP. Th ollowing thorm provs that th problm is NP-har. Thorm 4. It is NP-har to color an arbitrary gomtric thicknss-t graph with 4t 1 colors. Proo. I t = 1, thn coloring a planar graph (i.., t = 1) with 4t 1 = 3 colors is NP-har [17]. Assum now that t > 1. Givn an instanc I(G, t 1, 2(t 1)+ 1), whr G G t 1, w construct th corrsponing graph H(G, t). W prov that G is (2(t 1) + 1)-colorabl i an only i H(G, t) is (4t 1)-colorabl. I G is (2(t 1) + 1)-colorabl, thn w can color th copy o G that is contain in H(G, t) with 2(t 1) + 1 colors. Sinc thr os not xist any g in H(G, t) that conncts a vrtx o H i with a vrtx o H j, whr i j, w can color all H i s with 2t nw colors. Th rmaining vrtics (i.., th vrtics o H ) inuc a K 2t 1 such that non o ths vrtics ar ajacnt to th vrtics o G. Thror, w can rus th 2t 1 colors that w us to color th vrtics o G. Consquntly, numbr o colors w us to color H(G, t) is 2(t 1) t = 4t 1. On th othr han, assum that H(G, t) is (4t 1)-colorabl. Sinc th vrtics o H must hav 2t 1 irnt colors, an sinc ach H i is join with H, th vrtics o H i must us th rmaining 2t colors. Sinc vry vrtx v o G is join with a copy o istinct H i, v must b color with a color rom th 2t 1 colors us to color H. Thror, G must b (2t 1)-colorabl, i.., (2(t 1) + 1)-colorabl. 6. Discussion an Possibl Dirctions or Futur Rsarch In Sction 2 w construct n-vrtx gomtric thicknss-two graphs with 6n 19 gs, whr th bst known uppr boun on th numbr o gs is 6n 18 [16]. It is still unknown whthr thr xists a gomtric thicknss-two graph with 6n 18 gs. Opn Qustion 1. Dos thr xist a gomtric thicknss-two graph with n vrtics an 6n 18 gs? In Sction 3 w prov that any graph G with θ(g) > θ(g) must hav at last 10 vrtics. W construct such a graph G with 10 vrtics an 42 gs, whr θ(g) = 3 > θ(g) = 2. An intrsting qustion is whthr this inquality can b stablish or graphs with wr gs. Opn Qustion 2. Dos thr xist a graph G with wr than 42 gs satisying th inquality θ(g) > θ(g)? In Sction 4 w prov that rcognizing gomtric thicknss-two graphs is NP-har, which sttls an opn qustion pos in [6]. Thror, it sms natural to xamin whthr gomtric thicknss can b approximat icintly, a qustion also pos in [6, 13]. Not that it is possibl to trmin th thicknss within a constant actor [13]. 20

21 Figur 9: Two planar layrs o a 11-rgular 32-vrtx graph with thicknss two. Opn Qustion 3 ([6, 13]). Dos thr xist a constant-actor approximation algorithm to trmin gomtric thicknss? In Sction 5 w prov th NP-harnss o coloring arbitrary gomtric thicknss-t graphs with 4t 1 colors, which is particularly intrsting sinc no graph with gomtric thicknss t is known that rquirs mor than 4t colors or its propr coloring. Improving our complxity boun woul b an intrsting avnu to xplor. Opn Qustion 4. What is th complxity o coloring a gomtric thicknss-t graph with 4t colors? Dos vry gomtric thicknss-t graph amit a propr 4t-coloring? Evry planar graph contains a vrtx o gr at most iv, which is also th bst possibl sinc thr xists 5-rgular planar graphs. This uppr boun on th minimum gr o planar graphs las to a simpl algorithm or constructing 6-colorings o planar graphs [17]. Ringl [22] obsrv that th avrag gr o th graphs with thicknss-t is lss than 6t, which implis that vry 21

22 thicknss-t graph contains a vrtx o gr at most 6t 1. Consquntly, th chromatic numbr o gomtric thicknss-t graphs is 6t. For xampl, any graph G with θ(g) = 2 must contain a vrtx with gr at most 11, which las to an algorithm or constructing 12-colorings o thicknss-two graphs. In Figur 9 w show that this uppr boun o 11 on th smallst gr is tight by constructing an 11-rgular thicknss-two graph. Whil this articl was unr rviw, Duncan [9] prov that thr xist (6t 1)-rgular graphs with thicknss t. H also prov that thr xist 5t-rgular graphs with gomtric thicknss at most t. W bliv that or t 2, th uppr boun on th minimum gr o gomtric thicknss-t graphs is lss than th boun or thicknss-t graphs. Whil thr xist 11-rgular thicknss-two graphs, no gomtric thicknss-two graph is known with minimum gr gratr than 7. Opn Qustion 5. What is th smallst intgr k such that vry gomtric thicknss-t graph contains a vrtx o gr at most k? Acknowlgmnts. W thank th anonymous rviwrs or thir suggstions that hlp improv th prsntation o th papr. Rrncs [1] Aichholzr, O., Aurnhammr, F., Krassr, H.: Enumrating orr typs or small point sts with applications. Orr 19(3), (2002) [2] Alksv, V.B., Gonchakov, V.S.: Thicknss o arbitrary complt graphs. Math USSR Sbornik 30(2), (1976) [3] Appl, K., Hakn, W.: Evry planar map is our colorabl. Part I. Discharging. Illinois Journal o Mathmatics 21, (1977) [4] Barát, J., Matoušk, J., Woo, D.R.: Boun-gr graphs hav arbitrarily larg gomtric thicknss. Elctronic Journal o Combinatorics 13 (2006) [5] Bink, L.W., Harary, F., Moon, J.W.: On th thicknss o th complt bipartit graph. Math. Proc. o th Cambrig Philosophical Socity 60, 1 6 (1964) [6] Brannburg, F.J., Eppstin, D., Goorich, M.T., Kobourov, S.G., Liotta, G., Mutzl, P.: Slct opn problms in graph rawing. In: Procings o th Intrnational Symposium on Graph Drawing. LNCS, vol. 2912, pp Springr (2003) [7] Dillncourt, M.B., Eppstin, D., Hirschbrg, D.S.: Gomtric thicknss o complt graphs. Journal o Graph Algorithms an Applications 4(3), 5 17 (2000) 22

23 [8] Dujmovic, V., Woo, D.R.: Graph trwith an gomtric thicknss paramtrs. Discrt & Computational Gomtry 37(4), (2007) [9] Duncan, C.: Maximizing th gr o (gomtric) thicknss-t rgular graphs. In: Procings o th Intrnational Workshop on Graph Drawing & Ntwork Visualization. Springr, To Appar (2015) [10] Duncan, C.A.: On graph thicknss, gomtric thicknss, an sparator thorms. Computational Gomtry: Thory an Applications 44(2), (2011) [11] Duncan, C.A., Eppstin, D., Kobourov, S.G.: Th gomtric thicknss o low gr graphs. In: Procings o th Intrnational Symposium on Computational Gomtry. pp ACM (2004) [12] Durochr, S., Gthnr, E., Monal, D.: Thicknss an colorability o gomtric graphs. In: Procings o th Intrnational Workshop on Graph- Thortic Concpts in Computr Scinc. LNCS, vol. 8165, pp Springr (2013) [13] Eppstin, D.: Sparating thicknss rom gomtric thicknss. Contmporary Mathmatics 342, (2004) [14] Ertn, C., Kobourov, S.G.: Simultanous mbing o planar graphs with w bns. Journal Graph Algorithms an Applications 9(3), (2005) [15] Giacomo, E.D., Liotta, G.: Simultanous mbing o outrplanar graphs, paths, an cycls. Intrnational Journal o Computational Gomtry & Applications 17(2), (2007) [16] Hutchinson, J.P., Shrmr, T.C., Vinc, A.: On rprsntations o som thicknss-two graphs. Comp. Gom.: Thory an Applications 13(3), (1999) [17] Jnsn, T.R., Tot, B.: Graph coloring problms. Wily-Intrscinc Sris in Discrt Mathmatics an Optimization, John Wily & Sons Inc., Nw York (1995) [18] Kainn, P.C.: Thicknss an coarsnss o graphs. Abhanlungn aus m Mathmatischn Sminar r Univ. Hamburg 39(88 95) (1973) [19] Mansil, A.: Dtrmining th thicknss o a graph is NP-har. In: Mathmatical Procings o th Cambrig Philosophical Socity. vol. 93, pp (1983) [20] McGra, A.R.A., Zito, M.: Th complxity o th mpir colouring problm. Algorithmica 68(2), (2014) [21] Mutzl, P., Onthal, T., Scharbrot, M.: Th thicknss o graphs: A survy. Graphs Combin 14, (1998) 23

24 [22] Ringl, G.: Fabungsproblm au lachn un graphn. VEB Dutschr Vrlag r Wissnschatn (1950) [23] Tutt, W.T.: Th thicknss o a graph. Inag. Math 25, (1963) 24

25 Appnix A. In th proo o Thorm 2 w claim that at most six vrtics o Γ can b straight-lin visibl to a common point, whr Γ is a gomtric thicknss two rawing o K 9 minus an g. In this sction w giv a tail proo or this claim. Suppos or a contraiction that w can insrt a vrtx v such that it can b straight-lin visibl to svn istinct vrtics o Γ. Dlt all th black gs, i.., th gs common to both triangulations, rom Γ. Lt Γ not th rsulting rawing. Figurs 5(a) an (b) show th caniat rawings that ar obtain rom Figurs 3(a) an (c), rspctivly. W o not xamin Figur 3(b) sparatly sinc its clos rgions ar similar to that o Figur 3(a). In ach planar layr o Γ, v must li on som boun rgion or on th unboun rgion. Obsrv that th boun rgions in ach planar layr ar a collction o triangls an quarangls. I v lis intrior to an mpty triangl in ach planar layr, thn it can hav at most six ajacnt gs. Thror, v must li on th unboun rgion or on a quarangl o som planar layr. Each such rgion Q is unill in Figur 5. W xamin ach caniat rgion Q, an show that Q cannot contain any point v which is straight-lin visibl to svn istinct vrtics. W istinguish two main cass pning on whthr Q is a rgion in Figur 5(a) (Cas A), or Figur 5(b) (Cas B). Cas A (Q blongs to Figur 5(a)): W consir svral sub-cass, as ollows. Cas A1 (Q = ihg) : In this cas on can partition Q into 5 rgions Q 1, Q 2,..., Q 5, pning on how th nighboring sgmnts o h intrscts Q. Lt q b a point in Q. I q Q 1 Q 2, thn sinc {, g} an {i} li on th opposit sis o th lin trmin by sgmnt hb, vrtx g cannot b visibl to q. Thror, q can s only 3 points in th blu layr. Obsrv now that q is ithr insi a triangl in th r layr, or insi th quarangl ch. I q is insi a triangl in th r layr, thn q can s at most six vrtics in total, an hnc assum that q lis insi ch. Sinc {, h} an {, c} li on th opposit sis o th lin trmin by sgmnt ig, vrtx h cannot b visibl to q. Thror, also in this cas q can s at most six vrtics in total. Th cas whn q Q 4 Q 5 can b analyz in a similar ashion, whr an g cannot b visibl to q simultanously, an similarly h an cannot b visibl to q simultanously. In th rmaining cas w hav q Q 3, whr q can s all th vrtics {i, g, h, } in th blu layr. Sinc q is on th outr ac in th r layr, only th vrtics {b, c, i, h} ar visibl to q. Thror, q has straight-lin visibility to at most six points in total. Cas A2 (Q = ha) : In this cas v ithr lis in b Q, or in ch Q, or outsi o bc. I v b Q, thn v can s ithr {, b, g} or {, b, g} in th r layr. Sinc both o ths sts contain a vrtx common to {,, h, a}, vrtx v cannot 25

26 hav straight-lin visibility to mor than six istinct vrtics. I v ch Q, thn v can hav straight-lin visibility to on o th ollowing sts in th r layr: {, c, g}, {, b, g}, {, b, g}, {, g, c}, {,, h, c}, an {,, g}. Each o ths sts xcpt {,, h, c} contains a vrtx common to V (Q) = {,, h, a}, whras th st {,, h, c} contains two vrtics that ar common to V (Q) = {,, h, a}. Thror, vrtx v cannot hav straight-lin visibility to mor than six istinct vrtics. I v is outsi o bc, thn it is in th unboun ac in r layr. This cas is consir latr in Cas A5. Cas A3 (Q = hc) : Consir irst th cas whr v lis in ia Q. Hr v can hav straight-lin visibility to on o th ollowing sts in th blu layr: {,, i}, {i,, h, g}, {,, h, a}, {a, h, g}, {g, i, }, an {a, g, }. Each o ths sts o carinality thr has a vrtx common to V (Q) = {,, h, c}. Th st {,, h, a} has two vrtics common to V (Q) = {,, h, c}. Thror, v cannot s any o ths sts an also hav straight-lin visibility to mor than six istinct vrtics. Th rmaining st is {i,, h, g}, but by Cas A1, v cannot blong to th quarangl ihg. I v is outsi o ia, thn v is in th unboun ac in th blu layr. This cas is consir latr in Cas A4. Cas A4 (Q is th unboun ac in th blu layr): Lt H a b th hal-plan containing i, which is trmin by th lin through a an. Similarly, lt H b b th hal-plan containing i, which is trmin by th lin through b an. W istinguish two cass pning on whthr v H a H b or not. Consir irst th cas whn v H a H b. I v is also in th hal-plan H b, which contains {a,, g}, thn v cannot s h in any layr. Not that in this scnario v is also in th unboun ac o th r layr. Sinc thr ar only svn vrtics {a, b, c,,, h, i} in th unboun acs o th r an blu layrs, v cannot hav straight-lin visibility to mor than six vrtics. W may thus assum that v lis insi th quarangl bh. In this scnario v can s on o th ollowing sts in th r layr: {b,, h}, {b, h, c, i}, {h,,, c}. Th sts {b,, h} an {b, h, c, i} contains on an two vrtics that ar common to V (Q) = {a,, b, i}, rspctivly. Thror, v cannot s any o ths sts an also hav straight-lin visibility to mor than six istinct vrtics. Th rmaining st is {h,,, c}, whr V (Q). Thror, it suics to show that cannot b straight-lin visibl to v. Sinc v Q, it lis in bi. Not that bi bh bh. Thror, th g v must cross ithr g h or bh. Both h or bh blong to th r layr. Sinc is intrior to th blu layr, v must blong to th r layr, which implis that v cannot xist. Consir now th cas whn v is outsi o H a H b. I v is also on th unboun ac o th r layr, thn v hav to s all th svn vrtics {a, b, c,,, h, i}. Not that h an can b visibl to v only in th r layr. Thror, in this scnario v must li on th vrtically opposit angls o ch. Sinc i lis on th vrtically opposit angls o hc, th visibility rom v to h must b block by th g ci o th r layr. 26

27 I v is outsi o H a H b, but not on th unboun ac o th r layr, thn v can s on o th ollowing sts in th r layr: {, g, c}, {, g, c}, {,, h, c}. Sinc v Q, v can s only th vrtics {a,, b} in th blu layr. Thror, v can s at most six istinct vrtics altogthr. Cas A5 (Q is th unboun ac in th r layr): Lt H b b th hal-plan containing a, which is trmin by th lin through b an. Similarly, lt H c b th hal-plan containing a, which is trmin by th lin through an c. W istinguish two cass pning on whthr v H b H c or not. W can concntrat only on th cas whn v is not in th unboun ac o th blu layr (othrwis, w rr to Cas A4). I v H b H c, thn v can s on o th ollowing sts in th blu layr: {a, i, }, {a,,, h}, {a, h, g}, {a, g, }. Sinc abi ab abc, only th vrtics {b,, c} ar visibl to v in th r layr. Thror, at most six istinct vrtics can b visibl to v. I v is outsi o H b H c, thn v can s on o th ollowing sts in th blu layr: {a, i, }, {, i, }, {, i, g, h}, {i, g, }. Sinc bci bch bc, only th vrtics {b, h, c, i} ar visibl to v in th r layr. Thror, at most six istinct vrtics can b visibl to v. Cas B (Q blongs to Figur 5(b)): W consir svral sub-cass, as ollows. Cas B1 (Q = ig) : Not that bgc bc bc. Thror, i v gb, thn v can s on o th ollowing sts in th r layr: {b, c,, g}, {b,, h}, {b, i, h}. Th corrsponing point sts that ar visibl to v rom th blu layr ar {,, g, i}, {, g, i}, {, g, i}, rspctivly. Thror, v can s at most six istinct vrtics in total. Othrwis, i v g, thn v can s on o th ollowing sts in th r layr: {b, c,, g}, {c,, h}, {b,, h}, {b, i, h}, {i, h, c}, {i, c, }, {i,, }. For th st {b, c,, g}, th points that ar visibl in th blu layrs ar {i,, g, }, which implis visibility to only six istinct vrtics. For th rmaining sts, only thr mor points ar visibl in th blu layrs, i.., {, g, }. Cas B2 (Q = ah) : I v is in th unboun ac in th r layr, thn w rr to Cas B5. Othrwis, v can li in b or in ch. I v b, thn v can s on o th ollowing sts in th r layr: {b, i, }, {, i, }, {i, c,, }, {i, c, h}, {b, h, i}, {b,, h}. Sinc bc bc an, h to th opposit sis o th g g, v cannot s both an h simultanously in th blu layr. Thror, th st visibl to v in th blu layr is {a,, }. Hnc th numbr o istinct points visibl to v is at most six. I v ch, thn v can s on o th ollowing sts in th r layr: {c,, i, }, {, i, }, {i, c, h}. Sinc bc bc an, h to th opposit sis o th g g, v cannot s both an h simultanously in th blu layr. Thror, th st visibl to v in th blu layr is {a, h, }. Hnc th numbr o istinct points visibl to v is at most six. Cas B3 (Q = ic) : I v is in th unboun ac in th blu layr, thn w rr to Cas B4. Othrwis, v can li in a or in gi. 27

28 I v a, thn v can s on o th ollowing sts in th blu layr: {a, h, }, {a,,, h}. Sinc i ic, th st visibl to v in th r layr is {,, c}. Hnc th numbr o istinct points visibl to v is at most six. I v gi, thn v can s on o th ollowing sts in th blu layr: {, g, h}, {a,, h}, {a,,, h}, {, g, h}, {i, g,, }, {a, g, i}. Sinc i ic, th st visibl to v in th r layr is {i,, c}. Hnc th numbr o istinct points visibl to v is at most six, xcpt whn w tak th union o th sts {a,,, h} an {i,, c}. Howvr, by Cas B2, v cannot li in th quarangl ah. Cas B4 (Q is th unboun ac in th blu layr): Sinc th vrtics on th unboun ac o th blu layr ar {a, g,, b}, v cannot li insi any r triangl that is ajacnt to b or c. Thror, v must li in th quarangl ic or in th unboun ac o both r an blu layr. I v lis in th th quarangl ic, thn sinc ab abc, v can s only th vrtics {a, b, } in th blu layr, which implis that v is visibl to at most six istinct vrtics. Assum now that v is in th unboun ac o both r an blu layr. Thn to mak visibl ajacnt to svn vrtics, v must s all th vrtics {a, b, c,,,, g}. Sinc, ar intrnal vrtics in th blu layr, thy must b sn rom th r layr. Thror, v must li in th angl vrtically opposit to c. Not that bg bc, an is lis intrior to bg insi th gc. Thror, th visibility rom v to must b block by th g cg. Cas B5 (Q is th unboun ac in th r layr): Lt H b b th hal-plan containing a, which is trmin by th lin through b an. Similarly, lt H c b th hal-plan containing a, which is trmin by th lin through an c. W istinguish two cass pning on whthr v H b H c or not. W can concntrat only on th cas whn v is not in th unboun ac o th blu layr (othrwis, w rr to Cas B4). I v H b H c, thn v can s on o th ollowing sts in th blu layr: {a, i, g}, {a, i, }, {a,,, h}, {a, h, }. Sinc bcg bc bc, only th vrtics {b,, c} ar visibl to v in th r layr. Thror, at most six istinct vrtics can b visibl to v. I v is outsi o H b H c, thn v can s on o th ollowing sts in th blu layr: {a, i, g}, {i, g,, }, {g,, h}, {g, h, }, {g, h, }. Sinc bcg bc bc, only th vrtics {b, c,, g} ar visibl to v in th r layr. Thror, at most six istinct vrtics can b visibl to v. 28

29 Appnix B. Lt Γ b a thicknss-two rawing o K 9. Assum that th gs o on layr o Γ ar assign r color, an th rmaining gs ar assign blu color. Obsrv that ach o th unsaturat vrtics {, } o Γ is nclos by a cycl o istinct color, as ollows. I Γ corrspons to Figur 3(a) or (b), thn is nclos insi a blu cycl C() = (a,, i, g, a), whil is nclos insi a r cycl C() = (c,, h, c). I Γ corrspons to Figur 3(c), thn is nclos insi a r cycl C() = (c, i,, c), whil is insi a blu cycl C() = (a, i, g, h, a). Not that lis outsi o C(), whil lis outsi o C(). Furthrmor, an ar ajacnt to som vrtx o C() an C(), rspctivly. Thror, w can rprsnt ths conigurations as shown in Figurs B.1(a) (b). Lt [C()] an [C()] b th clos intrior o th cycl C() an C(), rspctivly. As illustrat in Figurs B.1(a) (b), w hav th ollowing obsrvation. Fact 2. Lt H b a graph obtain by aing a vrtx v to th unsaturat vrtics, o th graph K 9. Thn any thicknss-two rawing o graph H satisis th ollowing proprtis: - Th vrtx v lis ithr insi [C()] [C()], or outsi o [C()] [C()]. - Th gs (v, ) an (v, ) must li in irnt layrs. Lt G 1, G 2,..., G k b k 9 copis o K 9. Lt 1, 1 b th unsaturat vrtics o G 1. For ach G j, j > 1, mak 1 ajacnt to som unsaturat vrtx j o G j. Rr to th rmaining unsaturat vrtx o G j as j. A a vrtx v an mak v ajacnt to all th unsaturat vrtics o G 1, G 2,..., G k. Lt H k not th rsulting graph, which w rr to as a rigi graph. Lmma 1. Lt H k b a rigi graph. Thn in any thicknss-two rawing Γ o H k, th subgraph G inuc by th gs (v, 1 ), (v, j ) an ( 1, j ), whr 1 < j k, lis in th sam layr. Proo. To prov that th gs o G lis in th sam layr, w show that or vry j > 1, th gs o th triangl v, 1, j li in th sam layr. Lt D i b th rawing o G i in Γ. Consir th graphs G 1, G j. Sinc v can s both 1 an 1, by Fact 2, v lis ithr insi [C( 1 )] [C( 1 )] or outsi o [C( 1 )] [C( 1 )]. Similarly, sinc v can s both j an j, by Fact 2, v lis ithr insi [C( j )] [C( j )] or outsi o [C( j )] [C( j )]. Without loss o gnrality assum that C( 1 ) is a blu cycl in Γ. First assum that v lis insi [C( 1 )] [C( 1 )]. By Facts 1 an 2, th vrtics v, j an j ar connct both in r an blu layrs. Thror, j an j woul also li insi [C( 1 )] [C( 1 )], i.., {v, j, j } [C( 1 )] [C( 1 )]. W now consir Cass 1 2 pning on whthr C( j ) is blu or r. Cas 1 (C( j ) is blu): Sinc {v, j, j } [C( 1 )] [C( 1 )] an j is ajacnt to som vrtx o C( j ) through som blu g, th cycl C( j ) cannot nclos C( 1 ). Furthrmor, sinc th cycl C( 1 ) is blu an 1 29