Journl of Grph Algorithms n Applitions http://jg.info/ vol. 13, no. 2, pp. 153 177 (2009) On Clss of Plnr Grphs with Stright-Line Gri Drwings on Liner Are M. Rezul Krim 1,2 M. Siur Rhmn 1 1 Deprtment of Computer Siene n Engineering, Bnglesh University of Engineering n Tehnology (BUET), Dhk-1000, Bnglesh. 2 Deprtment of Computer Siene n Engineering, University of Dhk, Dhk-1000, Bnglesh. Astrt A stright-line gri rwing of plnr grph G is rwing of G on n integer gri suh tht eh vertex is rwn s gri point n eh ege is rwn s stright-line segment without ege rossings. It is well known tht plnr grph of n verties mits stright-line gri rwing on gri of re O(n 2 ). A lower oun of Ω(n 2 ) on the re-requirement for stright-line gri rwings of ertin plnr grphs re lso known. In this pper, we introue firly lrge lss of plnr grphs whih mits stright-line gri rwing on gri of re O(n). We give linertime lgorithm to fin suh rwing. Our new lss of plnr grphs, whih we ll oughnut grphs, is sulss of 5-onnete plnr grphs. We show severl interesting properties of oughnut grphs in this pper. One n esily oserve tht ny spnning sugrph of oughnut grph lso mits stright-line gri rwing with liner re. But the reognition of spnning sugrph of oughnut grph seems to e non-trivil prolem, sine the reognition of spnning sugrph of given grph is n NP-omplete prolem in generl. We estlish neessry n suffiient onition for 4-onnete plnr grph G to e spnning sugrph of oughnut grph. We lso give liner-time lgorithm to ugment 4-onnete plnr grph G to oughnut grph if G stisfies the neessry n suffiient onition. Sumitte: April 2008 Artile type: Regulr pper Reviewe: Jnury 2009 Finl: April 2009 Revise: Ferury 2009 Aepte: April 2009 Pulishe: June 2009 Communite y: Petr Mutzel E-mil resses: rkrim@univhk.eu (M. Rezul Krim) siurrhmn@se.uet.. (M. Siur Rhmn)
154 Krim n Rhmn Stright-Line Gri Drwings on Liner Are 1 Introution Reently utomti estheti rwings of grphs hve rete intense interest ue to their ro pplitions in omputer networks, VLSI lyout, informtion visuliztion et., n s onsequene numer of rwing styles hve ome out [4, 11, 14, 16]. A lssil n wiely stuie rwing style is the strightline rwing of plnr grph. A stright-line rwing of plnr grph G is rwing of G suh tht eh vertex is rwn s point n eh ege is rwn s stright-line segment without ege rossings. A stright-line gri rwing of plnr grph G is stright-line rwing of G on n integer gri suh tht eh vertex is rwn s gri point s shown in Figure 1(). h F 1 h j i p l k o g () F 2 m e n f f g i o n e j p m () k l F1 f g n h e m F 2 o i f g h i l p k () j e n o m p l () k j Figure 1: () A plnr grph G, () stright-line gri rwing of G with re O(n 2 ), () oughnut emeing of G n () stright-line gri rwing of G with re O(n). Wgner [19], Fry [6] n Stein [18] inepenently prove tht every plnr grph G hs stright-line rwing. Their proofs immeitely yiel polynomil-
JGAA, 13(2) 153 177 (2009) 155 time lgorithms to fin stright-line rwing of given plne grph. However, the re of retngle enlosing rwing on n integer gri otine y these lgorithms is not oune y ny polynomil of the numer n of verties in G. In ft, to otin rwing of re oune y polynomil remine s n open prolem for long time. In 1990, e Frysseix et l. [3] n Shnyer [17] showe y two ifferent methos tht every plnr grph of n 3 verties hs strightline rwing on n integer gri of size (2n 4) (n 2) n (n 2) (n 2), respetively. Figure 1() illustrtes stright-line gri rwing of the grph G in Figure 1() with re O(n 2 ). A nturl question rises: wht is the minimum size of gri require for stright-line rwing? e Frysseix et l. showe tht, for eh n 3, there exists plne grph of n verties, for exmple neste tringles, whih nees gri size of t lest 2(n 1)/3 2(n 1)/3 for ny gri rwing [2, 3]. Reently Frti n Ptrignni showe tht n 2 /9 + Ω(n) re is neessry for ny plnr stright-line rwing of neste tringles grph [7]. (Note tht plne grph is plnr grph with given emeing.) It hs een onjeture tht every plne grph of n verties hs gri rwing on 2n/3 2n/3 gri, ut it is still n open prolem. For some restrite lsses of grphs, more ompt stright-line gri rwings re known. For exmple, 4-onnete plne grph G hving t lest four verties on the outer fe hs stright-line gri rwing with re ( n/2 1) ( n/2 ) [15]. Grg n Rusu showe tht n n-noe inry tree hs plnr stright-line gri rwing with re O(n) [9]. Although trees mit stright-line gri rwings with liner re, it is generlly thought tht tringultions my require gri of qurti size. Hene fining nontrivil lsses of plnr grphs of n verties riher thn trees tht mit stright-line gri rwings with re o(n 2 ) is poste s n open prolem in [1]. Grg n Rusu showe tht n outerplnr grph with n verties n mximum egree hs plnr stright-line rwing with re O(n 1.48 ) [10]. Reently Di Bttist n Frti showe tht lne outerplnr grph of n verties hs stright-line gri rwing with re O(n) n generl outerplnr grph of n verties hs stright-line gri rwing with re O(n 1.48 ) [5]. In this pper, we introue new lss of plnr grphs whih hs strightline gri rwing on gri of re O(n). We give liner-time lgorithm to fin suh rwing. Our new lss of plnr grphs is sulss of 5-onnete plnr grphs, n we ll the lss oughnut grphs sine grph in this lss hs oughnut-like emeing s illustrte in Figure 1(). In n emeing of oughnut grph of n verties, there re two vertex-isjoint fes eh hving extly n/4 verties n eh of ll the other fes hs extly three verties. Figure 1() illustrtes oughnut grph of 16 verties where eh of the two fes F 1 n F 2 ontins four verties n eh of ll other fes ontins extly three verties. Figure 1() illustrtes oughnut-like emeing of G where F 1 is emee s the outer fe n F 2 is emee s n inner fe. A stright-line gri rwing of G with re O(n) is illustrte in Figure 1(). The outerplnrity of oughnut grph is 3. Thus oughnut grphs introue sulss of 3-outerplnr grphs tht mits stright-line gri rwing with liner re. One n esily oserve tht ny spnning sugrph of ough-
156 Krim n Rhmn Stright-Line Gri Drwings on Liner Are nut grph lso mits stright-line gri rwing with liner re. But the reognition of spnning sugrph of given grph is n NP-omplete prolem in generl. We estlish neessry n suffiient onition for 4-onnete plnr grph to e spnning sugrph of oughnut grph. We lso provie liner-time lgorithm to ugment 4-onnete grph G to oughnut grph if G stisfies the neessry n suffiient onition. This gives us new lss of grphs whih is sulss of 4-onnete plnr grphs tht mits stright-line gri rwings with liner re. The reminer of the pper is orgnize s follows. In Setion 2, we give some efinitions. Setion 3 provies some properties of the lss of oughnut grphs. Setion 4 els with stright-line gri rwings of oughnut grphs. Setion 5 provies the hrteriztion for 4-onnete plnr grph to e spnning sugrph of oughnut grph. Finlly Setion 6 onlues the pper. Erly versions of this pper hve een presente t [12] n [13]. 2 Preliminries In this setion we give some efinitions. Let G = (V, E) e onnete simple grph with vertex set V n ege set E. Throughout the pper, we enote y n the numer of verties in G, tht is, n = V, n enote y m the numer of eges in G, tht is, m= E. An ege joining verties u n v is enote y (u, v). The egree of vertex v, enote y (v), is the numer of eges inient to v in G. G is lle r-regulr if every vertex of G hs egree r. We ll vertex v neighor of vertex u in G if G hs n ege (u, v). The onnetivity κ(g) of grph G is the minimum numer of verties whose removl results in isonnete grph or single-vertex grph K 1. G is lle k-onnete if κ(g) k. We ll vertex of G ut-vertex of G if its removl results in isonnete or single-vertex grph. For W V, we enote y G W the grph otine from G y eleting ll verties in W n ll eges inient to them. A ut-set of G is set S V (G) suh tht G S hs more thn one omponent or G S is single vertex grph. A pth in G is n orere list of istint verties v 1, v 2,..., v q V suh tht (v i 1, v i ) E for ll 2 i q. Verties v 1 n v q re en-verties of the pth v 1, v 2,..., v q. Two pths re vertex-isjoint if they o not shre ny ommon vertex exept their en verties. The length of pth is the numer of eges on the pth. We ll pth P n even pth if the numer of eges on P is even. We ll pth P n o pth if the numer of eges on P is o. A grph is plnr if it n e emee in the plne so tht no two eges interset geometrilly exept t vertex to whih they re oth inient. A plne grph is plnr grph with fixe emeing. A plne grph G ivies the plne into onnete regions lle f es. A oune region is lle n inner fe n the unoune region is lle the outer fe. For fe F in G we enote y V (F ) the set of verties of G on the ounry of fe F. Two fes F 1 n F 2 re vertex-isjoint if V (F 1 ) V (F 2 ) =. Let F e fe in plne grph G with n 3. If the ounry of F hs extly three verties
JGAA, 13(2) 153 177 (2009) 157 then we ll F tringulte fe. One n ivie fe F of p (p 3) verties into p 2 tringulte fes y ing p 3 extr eges. The opertion ove is lle tringulting fe. If every fe of grph is tringulte, then the grph is lle tringulte plne grph. We n otin tringulte plne grph G from non-tringulte plne grph G y tringulting ll fes of G. A mximl plnr grph is one to whih no ege n e e without losing plnrity. Thus the ounry of every fe of G is tringle in ny emeing of mximl plnr grph G with n 3, n hene n emeing of mximl plnr grph is often lle tringulte plne grph. It n e erive from Euler s formul for plnr grphs tht if G is mximl plnr grph with n verties n m eges then m = 3n 6, for more etils see [16]. We ll fe qurngle fe if the fe hs extly four verties. For ny 3-onnete plnr grph the following ft hols. Ft 1 Let G e 3-onnete plnr grph n let Γ n Γ e ny two plnr emeings of G. Then ny fil yle of Γ is fil yle of Γ n vie vers. Let G e 5-onnete plnr grph, let Γ e ny plnr emeing of G n let p e n integer suh tht p 4. We ll G p-oughnut grph if the following onitions ( 1 ) n ( 2 ) hol: ( 1 ) Γ hs two vertex-isjoint fes eh of whih hs extly p verties, n ll the other fes of Γ hs extly three verties; n ( 2 ) G hs the minimum numer of verties stisfying onition ( 1 ). In generl, we ll p-oughnut grph for p 4 oughnut grph. Sine oughnut grph is 5-onnete plnr grph, Ft 1 implies tht the eomposition of oughnut grph into its fil yles is unique. Throughout the pper we often mention fes of oughnut grph G without mentioning its plnr emeing where the esription of the fes is vli for ny plnr emeing of G. 3 Properties of Doughnut Grphs In this setion we will show some properties of p-oughnut grph. We hve the following lemm on the numer of verties of grph stisfying onition ( 1 ). Lemm 2 Let G e 5-onnete plnr grph, let Γ e ny plnr emeing of G, n let p e n integer suh tht p 4. Assume tht Γ hs two vertexisjoint fes eh of whih hs extly p verties, n ll the other fes of Γ hs extly three verties. Then G hs t lest 4p verties. Proof: Let F 1 n F 2 e the two fes of Γ eh of whih ontins extly p verties. Let x e the numer of verties in G whih re neither on F 1 nor on F 2. Then G hs x + 2p verties.
158 Krim n Rhmn Stright-Line Gri Drwings on Liner Are We lulte the numer of eges in G s follows. Fes F 1 n F 2 of Γ re not tringulte sine p 4. If we tringulte F 1 n F 2 of Γ then the resulting grph G is mximl plnr grph. Using Euler s formul, G hs extly 3(x + 2p) 6 = 3x + 6p 6 eges. To tringulte eh of F 1 n F 2, we nee to p 3 eges n hene the numer of eges in G is extly (3x + 6p 6) 2(p 3) = 3x + 4p. (1) Sine G is 5-onnete, using the egree-sum formul, we get 2(3x + 4p) 5(x + 2p). This reltion implies x 2p. (2) Therefore G hs t lest 4p verties. Lemm 2 implies tht p-oughnut grph hs 4p or more verties. We now show tht 4p verties re suffiient to onstrut p-oughnut grph s in the following lemm. Lemm 3 For n integer p, p 4, one n onstrut p-oughnut grph G with 4p verties. To prove Lemm 3 we first onstrut plnr emeing Γ of G with 4p verties y the onstrution Construt-Doughnut given elow n then show tht G is p-oughnut grph. Construt-Doughnut. Let C 1, C 2, C 3 e three vertex-isjoint yles suh tht C 1 ontins p verties, C 2 ontins 2p verties n C 3 ontins p verties. Let x 1, x 2,..., x p e the verties on C 1, y 1, y 2,..., y p e the verties on C 3, n z 1, z 2,.., z 2p e the verties on C 2. Let R 1, R 2 n R 3 e three onentri irles on plne with rius r 1, r 2 n r 3, respetively, suh tht r 1 > r 2 > r 3. We eme C 1, C 2 n C 3 on R 1, R 2 n R 3 respetively, s follows. We put the verties x 1, x 2...x p of C 1 on R 1 in lokwise orer suh tht x 1 is put on the leftmost position mong the verties x 1, x 2,..., x p. Similrly, we put verties z 1, z 2,..., z 2p of C 2 on R 2 n y 1, y 2,..., y p of C 3 on R 3. We eges etween the verties on C 1 n C 2, n etween the verties on C 2 n C 3 s follows. We hve two ses to onsier. Cse 1: k is even in z k. In this se, we two eges (z k, x k/2 ), (z k, x i ) etween C 2 n C 1, n one ege (z k, y i ) etween C 2 n C 3 where i = 1 if k = 2p, n i = k/2 + 1 otherwise. Cse 2: k is o in z k. In this se, we two eges (z k, y k/2 ), (z k, y i ) etween C 2 n C 3, n one ege (z k, x k/2 ) etween C 1 n C 2 where i = 1 if k = 2p 1, n i = k/2 + 1 otherwise. We thus onstrute plnr emeing Γ of G. Figure 2 illustrtes the onstrution ove for the se of p = 4. We hve the following lemm on the onstrution Construt-Doughnut.
JGAA, 13(2) 153 177 (2009) 159 x z 1 1 z 8 y 1 x4 z 1 7 2 z6 y 4 3 z y 5 3 x3 x 1 x4 1 z7 z 8 2 z6 y 4 3 z 1 z y 1 y 5 3 x3 x 1 z 1 x4 1 z7 z 8 2 z6 y 4 3 z y 1 y 5 3 x3 z 2 y2 z3 x2 () z4 z 2 y2 z 3 x2 () z 4 z 2 y2 z 3 x2 () z 4 Figure 2: Illustrtion for the onstrution of plnr emeing Γ of p- oughnut grph G for p = 4; () emeing of the three yles C 1, C 2 n C 3 on three onentri irles, () ition of eges for the se where k is even in z k n () Γ. Lemm 4 Let Γ e the plne grph of 4p verties otine y the onstrution Construt-Doughnut. Then Γ hs extly two vertex-isjoint fes F 1 n F 2 eh of whih hs extly p verties, n the rest of the fes re tringulte. Proof: The onstrution of Γ implies tht yle C 1 is the ounry of the outer fe F 1 of Γ n yle C 3 is the ounry of n inner fe F 2. Eh of F 1 n F 2 hs extly p verties. Clerly the two fes F 1 n F 2 of Γ re vertex-isjoint. Thus it is remine to show tht the rest of the fes of Γ re tringulte. The rest of the fes n e ivie into two groups; (i) fes hving verties on oth the yles C 1 n C 2, n (ii) fes hving the verties on oth the yles C 2 n C 3. We only prove tht eh fe in group (i) is tringulte, sine the proof for group (ii) is similr. From our onstrution eh vertex z i with even i hs extly two neighors on C 1, n the two neighors of z i on C 1 re onseutive. Hene we get tringulte fe for eh z i with even i whih ontins z i n the two neighors of z i on C 1. We now show tht the remining fes in group (i) re tringulte. Clerly eh of the remining fes in group (i) must ontin vertex z i with o i sine vertex on fe in group (i) is either on C 1 or on C 2 n vertex on C 2 hs t most two neighors on C 1. Let z i, z i+1 n z i+2 e three onseutive verties on C 2 with even i. Then z i n z i+2 hs ommon neighor x on C 1. One n oserve from our onstrution tht x is lso the only neighor of z i+1 on C 1. Then extly two fes in group (i) ontin z i+1 n the two fes re tringulte. This implies tht for eh z i on C 2 with o i there re extly two fes in group (i) whih ontin z i, n the two fes re tringulte.
160 Krim n Rhmn Stright-Line Gri Drwings on Liner Are We re rey to prove Lemm 3. Proof of Lemm 3 We onstrut plnr emeing Γ of grph G with 4p verties y the onstrution Construt-Doughnut n show tht G is p-oughnut grph. To prove this lim we nee to prove tht G stisfies the following properties () (): () the grph G is 5-onnete plnr grph; () ny plnr emeing Γ of G hs extly two vertex-isjoint fes eh of whih hs extly p verties, n ll the other fes re tringulte; n () G hs the minimum numer of verties stisfying () n (). () G is plnr grph sine it hs plnr emeing Γ s illustrte in Figure 2(). To prove tht G is 5-onnete, we show tht the size of ny ut-set of G is 5 or more. We first show tht G is 5-regulr. From the onstrution, one n esily see tht eh of the vertex of C 2 hs extly three neighors in V (C 1 ) V (C 3 ). Hene the egree of eh vertex of C 2 is extly 5. We only prove tht the egree of eh vertex of C 1 is extly 5 sine the proof is similr for the verties of C 3. Eh even inex vertex v of C 2 hs two neighors on C 1 n the two neighors of v re onseutive on C 1 y onstrution. Eh vertex u of C 1 hs t most two even inex neighors on C 2, sine C 2 hs p even inex verties, C 1 hs p verties, n Γ is plnr emeing. Assume tht vertex u of C 1 hs two even inex neighors y i n y i+2 on C 2. Sine Γ is plnr emeing y i+1 n hve only one neighor on C 1 whih is u. Thus vertex u on C 1 hs t most three neighors on C 2. Sine there re extly 3p eges eh of whih hs one en point on C 1 n the other on C 2, n vertex on C 1 hs t most three neighors on C 2, eh vertex of C 1 hs extly three neighors on C 2. Hene the egree of vertex on C 1 is 5. Therefore G is 5-regulr. We next show tht the size of ny ut-set of G is 5 or more. Assume for ontrition tht G hs ut-set of less thn five verties. In suh se, G woul hve vertex of egree less thn five, ontrition. (Note tht G is 5-regulr, the verties of G lie on three vertex isjoint yles C 1, C 2 n C 3, none of the verties of C 1 hs neighor on C 3, eh of the fes of G is tringulte exept fes F 1 n F 2.) () By Lemm 4, G hs plnr emeing Γ suh tht Γ hs extly two vertex-isjoint fes F 1 n F 2 eh of whih hs extly p verties, n the rest of the fes re tringulte. Sine G is 5-onnete, Ft 1 implies tht ny plnr emeing Γ of G hs extly two vertex-isjoint fes eh of whih hs extly p verties, n ll the other fes re tringulte. () We hve onstrute the grph G with 4p verties n proofs for () n () imply tht G stisfies properties () n (). G is 5-onnete plnr grph n hene stisfies onition ( 1 ) of the efinition of p-oughnut grph. By Lemm 2, 4p is the minimum numer of verties of suh grph.
JGAA, 13(2) 153 177 (2009) 161 Conition ( 2 ) of the efinition of p-oughnut grph n Lemms 2 n 3 imply tht p-oughnut grph G hs extly 4p verties. Then the vlue of x in Eq. (2) is 2p in G. By Eq. (1), G hs extly 3x + 4p = 10p eges. Sine G is 5-onnete, every vertex hs egree 5 or more. Then the egree-sum formul implies tht every vertex of G hs egree extly 5. Thus the following theorem hols. Theorem 1 Let G e p-oughnut grph. Then G is 5-regulr n hs extly 4p verties. For yle C in plne grph G, we enote y G(C) the plne sugrph of G insie C exluing C. Let C 1, C 2 n C 3 e three vertex-isjoint yles in plnr grph G suh tht V (C 1 ) V (C 2 ) V (C 3 )=V (G). Then we ll plnr emeing Γ of G oughnut emeing of G if C 1 is the outer fe n C 3 is n inner fe of Γ, G(C 1 ) ontins C 2 n G(C 2 ) ontins C 3. We ll C 1 the outer yle, C 2 the mile yle n C 3 the inner yle of Γ. We next show tht p-oughnut grph hs oughnut emeing. To prove the lim we nee the following lemms. Lemm 5 Let G e p-oughnut grph. Let F 1 n F 2 e the two fes of G eh of whih ontins extly p verties. Then G {V (F 1 ) V (F 2 )} is onnete n ontins yle. Proof: Sine G is 5-onnete, G = G {V (F 1 ) V (F 2 )} is onnete; otherwise, G woul hve ut-set of 4 verties - two of them re on F 1 n the other two re on F 2, ontrition. Clerly G hs extly 2p verties. Sine G is 5-regulr n hs extly 4p verties y Theorem 1, one n oserve following the egree-sum formul tht G ontins t lest 2p eges; if there is no ege etween vertex of F 1 n vertex of F 2 in G then G ontins extly 2p eges, otherwise G ontins more thn 2p eges. Sine G is onnete, hs 2p verties n hs t lest 2p eges, G must hve yle. Lemm 6 Let G e p-oughnut grph. Let F 1 n F 2 e the two fes of G eh of whih ontins extly p verties. Let Γ e plnr emeing of G suh tht F 1 is emee s the outer fe. Let C e yle in G {V (F 1 ) V (F 2 )}. Then G(C) in Γ ontins F 2. Proof: Assume tht G(C) oes not ontin F 2 in Γ. Sine F 1 is emee s the outer fe of Γ, F 2 will e n inner fe of Γ s illustrte in Figure 3. Then there woul e ege rossings in Γ mong the eges from the verties on C to the verties on F 1 n F 2 s illustrte in Figure 3, ontrition to the ssumption tht Γ is plnr emeing of G. (Note tht G is 5-onnete, 5-regulr n hs 10p eges.) Therefore G(C) ontins F 2. Lemm 7 Let G e p-oughnut grph. Let F 1 n F 2 e the two fes of G eh of whih ontins extly p verties. Then the following () - () hol. () There is no ege etween vertex of F 1 n vertex of F 2.
162 Krim n Rhmn Stright-Line Gri Drwings on Liner Are F 1 C F 2 Figure 3: An emeing Γ of G where F 1 is emee s the outer fe n G(C ) oes not ontin F 2. () G {V (F 1 ) V (F 2 )} ontins extly 2p eges n hs extly one yle. () All the verties of G {V (F 1 ) V (F 2 )} re ontine in single yle. Proof: () Let Γ e plnr emeing of G suh tht F 1 is emee s the outer fe. Let C e yle in G {V (F 1 ) V (F 2 )}. Sine G(C) ontins F 2 y Lemm 6, there is no ege etween vertex of F 1 n vertex of F 2 ; otherwise, n ege etween vertex of F 1 n vertex of F 2 woul ross n ege on C, ontrition to the ssumption tht Γ is plnr emeing of G. () Sine there is no ege etween vertex of F 1 n vertex of F 2 y Lemm 7(), G {V (F 1 ) V (F 2 )} ontins extly 2p eges s mentione in the proof of Lemm 5. Sine G {V (F 1 ) V (F 2 )} is onnete, ontins extly 2p verties n hs extly 2p eges, G {V (F 1 ) V (F 2 )} ontins extly one yle. () Let Γ e plnr emeing of G suh tht F 1 is emee s the outer fe. Let C e yle in G {V (F 1 ) V (F 2 )}. By Lemm 7(), C is the only yle in G {V (F 1 ) V (F 2 )}. Assume tht the yle C oes not ontin ll the verties of G {V (F 1 ) V (F 2 )}. Then there is t lest vertex in G {V (F 1 ) V (F 2 )} whose egree is one in G {V (F 1 ) V (F 2 )}. Let v e vertex of egree one in G {V (F 1 ) V (F 2 )}. We my ssume tht the vertex v is outsie of the yle C in Γ sine the proof is similr if v is insie of the yle C. Then the four neighors of v must e on F 1, sine G {V (F 1 ) V (F 2 )} ontins extly one yle y Lemm 7(), the vertex v is outsie of the yle C, n G(C) ontins F 2 y Lemm 6. Then either G woul not e 5-regulr or Γ woul not e plnr emeing of G, ontrition. Hene ll the verties of G {V (F 1 ) V (F 2 )} re ontine in yle C. We now prove the following theorem. Theorem 2 A p-oughnut grph lwys hs oughnut emeing.
JGAA, 13(2) 153 177 (2009) 163 Proof: Let F 1 n F 2 e two fes of G eh of whih ontins extly p verties. Let Γ e plnr emeing of G suh tht F 1 is emee s the outer fe. By Lemm 7(), ll the verties of G {V (F 1 ) V (F 2 )} re ontine in single yle C. By Lemm 6, G(C) ontins F 2. Then in Γ, G(F 1 ) ontins C n G(C) ontins F 2, n hene Γ is oughnut emeing. A 1-outerplnr grph is n emee plnr grph where ll verties re on the outer fe. It is lso lle 1-outerplne grph. An emee grph is k-outerplne (k > 1) if the emee grph otine y removing ll verties of the outer fe is (k 1)-outerplne grph. A grph is k-outerplnr if it mits k-outerplnr emeing. A plnr grph G hs outerplnrity k (k > 0) if it is k-outerplnr n it is not j-outerplnr for 0 < j < k. In the rest of this setion, we show tht the outerplnrity of p-oughnut grph G is 3. Sine none of the fes of G ontins ll verties of G, G oes not mit 1-outerplnr emeing. We thus nee to show tht G oes not mit 2-outerplnr emeing. We hve the following ft. Ft 8 A grph G hving outerplnrity 2 hs ut-set of four or less verties. Proof: Deletion of ll verties on the outer fe from 2-outerplne grph leves 1-outerplne grph. Sine ll verties of 1-outerplne grph re on the outer fe, 1-outerplne grph hs ut-set of t most two verties. Then one n oserve tht grph G hving outerplnrity 2 hs ut-set of four or less verties. Sine G is 5-onnete grph, G hs no ut-set of four or less verties. Hene y Ft 8 the grph G hs outerplnrity greter thn 2. Thus the following lemm hols. Lemm 9 Let G e p-oughnut grph for p 4. Then G is neither 1- outerplnr grph nor 2-outerplnr grph. We now prove the following theorem. Theorem 3 The outerplnrity of p-oughnut grph G is 3. Proof: A oughnut emeing of G immeitely implies tht G hs 3- outerplnr emeing. By Lemm 9, G is neither 1-outerplnr grph nor 2-outerplnr grph. Therefore the outerplnrity of p-oughnut grph is 3. 4 Drwings of Doughnut Grphs In this setion we give liner-time lgorithm for fining stright-line gri rwing of oughnut grph on gri of liner re. Let G e p-oughnut grph. Then G hs oughnut emeing y Theorem 2. Let Γ e oughnut emeing of G s illustrte in Figure 4(). Let C 1, C 2 n C 3 e the outer yle, the mile yle n the inner yle of Γ, respetively. We hve the following fts.
164 Krim n Rhmn Stright-Line Gri Drwings on Liner Are Ft 10 Let G e p-oughnut grph n let Γ e oughnut emeing of G. Let C 1, C 2 n C 3 e the outer yle, the mile yle n the inner yle of Γ, respetively. For ny two onseutive verties z i, z i+1 on C 2, one of z i, z i+1 hs extly one neighor on C 1 n the other hs extly two neighors on C 1. Ft 11 Let G e p-oughnut grph n let Γ e oughnut emeing of G. Let C 1, C 2 n C 3 e the outer yle, the mile yle n the inner yle of Γ, respetively. Let z i e vertex of C 2, then either the following () or () hols. () z i hs extly one neighor on C 1 n extly two neighors on C 3. () z i hs extly one neighor on C 3 n extly two neighors on C 1. Before esriing our lgorithm we nee some efinitions. Let z i e vertex of C 2 suh tht z i hs two neighors on C 1. Let x n x e the two neighors of z i on C 1 suh tht x is the ounter lokwise next vertex to x on C 1. We ll x the left neighor of z i on C 1 n x the right neighor of z i on C 1. Similrly we efine the left neighor n the right neighor of z i on C 3 if vertex z i on C 2 hs two neighors on C 3. We re now rey to esrie our lgorithm. We eme C 1, C 2 n C 3 on three neste retngles R 1, R 2 n R 3, respetively on gri s illustrte in Figure 4(). We rw retngle R 1 on gri with four orners on gri point (0, 0), (p + 1, 0), (p + 1, 5) n (0, 5). Similrly the four orners of R 2 re (1, 1), (p, 1), (p, 4), (1, 4) n the four orners of R 3 re (2, 2), (p 1, 2), (p 1, 3), (2, 3). We first eme C 2 on R 2 s follows. Let z 1, z 2,..., z 2p e the verties on C 2 in ounter lokwise orer suh tht z 1 hs extly one neighor on C 1. We put z 1 on (1, 1), z p on (p, 1), z p+1 on (p, 4) n z 2p on (1, 4). We put the other verties of C 2 on gri points of R 2 preserving the reltive positions of verties of C 2. We next put verties of C 1 on R 1 s follows. Let x 1 e the neighor of z 1 on C 1 n let x 1, x 2,..., x p e the verties of C 1 in ounter lokwise orer. We put x 1 on (0, 0) n x p on (0, 5). Sine z 1 hs extly one neighor on C 1, y Ft 10, z 2p hs extly two neighors on C 1. Sine z 1 n z 2p re on tringulte fe of G hving verties on oth C 1 n C 2, x 1 is neighor of z 2p. One n esily oserve tht x p is the other neighor of z 2p on C 1. Clerly the eges (x 1, z 1 ), (x 1, z 2p ), (x p, z 2p ) n e rwn s stright-line segments without ege rossings s illustrte in Figure 4(). We next put neighors of z p n z p+1. Let x i e the neighor of z p on C 1 if z p hs extly one neighor on C 1, otherwise let x i e the left neighor of z p on C 1. We put x i on (p+1, 0) n x i+1 on (p+1, 5). In se of z p hs extly one neighor on C 1, y Ft 10, z p+1 hs two neighors on C 1, n x i n x i+1 re the two neighors of z p+1 on C 1. Clerly the eges (z p, x i ), (z p+1, x i ) n (z p+1, x i+1 ) n e rwn s strightline segments without ege rossings, s illustrte in Figure 4(). In se of z p hs extly two neighors x i n x i+1 on C 1, the eges etween neighors of z p n z p+1 on C 1 n e rwn without ege rossings s illustrte in Figure 5. We put the other verties of C 1 on gri points of R 1 ritrrily preserving their reltive positions on C 1.
JGAA, 13(2) 153 177 (2009) 165 C 1 x p C 2 z 2 p y p C 3 x p R 1 (0,5) ( p+1,5) x i+1 R2 z 2 p z (1,4) ( p,4) p+1 (2,3) (2,2) R 3 x1 z1 (p 1,3) (p 1,2) z2 y 1 z3 x2 y2 y3 z4 () x p z5 z 2 p x3 z p (0,5) ( p+1,5) (1,4) (2,3) (2,2) (p 1,3) (p 1,2) p,4) ( z p+1 x i+1 z 1 (1,1) ( p,1) z p z 1 (1,1) ( p,1) z p x 1 x p x i x 1 (0,0) ( p+1,0) (0,0) () (0,5) ( p+1,5) x i+1 x p () ( p+1,0) (0,5) ( p+1,5) xi x i+1 z 2 p z p+1 z 2 p z p+1 z 1 y p (2,3) (2,2) y 1 (p 1,3) (p 1,2) y i y i+1 zp z 1 y p y 1 (2,3) (2,2) (p 1,3) (p 1,2) y i y i+1 z p x 1 (0,0) () ( p+1,0) xi x 1 (0,0) (e) ( p+1,0) x i Figure 4: () A oughnut emeing of p-oughnut grph of G, () eges etween four orner verties of R 1 n R 2 re rwn s stright-line segments, () eges etween verties on R 1 n R 2 re rwn, () eges etween four orner verties of R 2 n R 3 re rwn s stright-line segments, n (e) stright-line gri rwing of G.
166 Krim n Rhmn Stright-Line Gri Drwings on Liner Are x p (0,5) R 1 ( p+1,5) R p+1 i+1 2 z x z 2 p (1,4) ( p,4) R 3 (2,3) (p 1,3) (2,2) (p 1,2) z 1 (1,1) ( p,1) z p x 1 (0,0) ( p+1,0) x i Figure 5: Illustrtion for the se where z p hs two neighors on C 1. One n oserve tht ll the eges of G onneting verties in {z 2, z 3,..., z p 1 } to verties in {x 2, x 3,.., x i 1 }, n onneting verties in {z p+2, z p+2,..., z 2p 1 } to verties in {x i+2, x i+3,.., x p 1 } n e rwn s stright-line segments without ege rossings. See Figure 4(). We finlly put the verties of C 3 on R 3 s follows. Sine z 1 hs extly one neighor on C 1, y Ft 11(), z 1 hs extly two neighors on C 3. Then, y Ft 11(), z 2p hs extly one neighor on C 3. Let y 1, y 2,..., y p e the verties on C 3 in ounter lokwise orer suh tht y 1 is the right neighor of z 1. Then y p is the left neighor of z 1. We put y 1 on (2, 2) n y p on (2, 3). Clerly the eges (y 1, z 1 ), (y p, z 2p ), (y p, z 1 ) n e rwn s stright-line segments without ege rossings, s illustrte in Figure 4(). We next put neighors of z p n z p+1 on C 3 s we hve put the neighors of z p n z p+1 on C 1 t the other two orners of R 3 in ounter lokwise orer s illustrte in Figure 4(). We put the other verties of C 3 on gri points of R 3 ritrrily preserving their reltive positions on C 3. It is not iffiult to show tht eges from the verties on C 2 to the verties on C 3 n e rwn s stright-line segments without ege rossings. Figure 4(e) illustrtes the omplete stright-line gri rwing of p-oughnut grph. The re requirement of the stright-line gri rwing of p-oughnut grph G is equl to the re of retngle R 1 n the re of R 1 is = (p + 1) 5 = (n/4 + 1) 5 = O(n), where n is the numer of verties in G. Thus we hve stright-line gri rwing of p-oughnut grph on gri of liner re. Clerly the lgorithm tkes liner time. Thus the following theorem hols. Theorem 4 A oughnut grph G of n verties hs stright-line gri rwing on gri of re O(n). Furthermore, the rwing of G n e foun in liner time.
JGAA, 13(2) 153 177 (2009) 167 5 Spnning Sugrphs of Doughnut Grphs In Setion 4, we hve seen tht oughnut grph mits stright-line gri rwing with liner re. One n esily oserve tht spnning sugrph of oughnut grph lso mits stright-line gri rwing with liner re. Figure 6() illustrtes stright-line gri rwing with liner re of grph G in Figure 6() where G is spnning sugrph of oughnut grph G in Figure 1(). Using trnsformtion from the sugrph isomorphism prolem [8], one n esily prove tht the reognition of spnning sugrph of given grph is n NP-omplete prolem in generl. Hene the reognition of spnning sugrph of oughnut grph seems to e non-trivil prolem. We thus restrit ourselves only to 4-onnete plnr grphs. In this setion, we give neessry n suffiient onition for 4-onnete plnr grph to e spnning sugrph of oughnut grph s in the following theorem. l F 1 k j p F i 2 o h g () m e n f f e g l h n o m p () k i j Figure 6: () A spnning sugrph G of G in Figure 1(), n () stright-line gri rwing of G with re O(n). Theorem 5 Let G e 4-onnete plnr grph with 4p verties where p > 4 n let (G) 5. Let Γ e plnr emeing of G. Assume tht Γ hs extly two vertex isjoint fes F 1 n F 2 eh of whih hs extly p verties. Then G is spnning sugrph of p-oughnut grph if n only if the following onitions () (e) hol. () G hs no ege (x, y) suh tht x V (F 1 ) n y V (F 2 ). () Every fe f of Γ hs t lest one vertex v {V (F 1 ) V (F 2 )}. () For ny vertex x / {V (F 1 ) V (F 2 )}, the totl numer of neighors of x on fes F 1 n F 2 re t most three. () Every fe f of Γ exept the fes F 1 n F 2 hs either three or four verties. (e) For ny x-y pth P suh tht V (P ) {V (F 1 ) V (F 2 )} = n x hs extly two neighors on fe F 1 (F 2 ). Then the following onitions hol. (i) If P is even, then the vertex y hs t most two neighors on fe F 1 (F 2 ) n t most one neighor on fe F 2 (F 1 ).
168 Krim n Rhmn Stright-Line Gri Drwings on Liner Are (ii) If P is o, then the vertex y hs t most one neighor on fe F 1 (F 2 ) n t most two neighors on fe F 2 (F 1 ). Ft 1 implies tht the eomposition of 4-onnete plnr grph G into its fil yles is unique. Throughout the setion we thus often mention fes of G without mentioning its plnr emeing where the esription of the fes is vli for ny plnr emeing of G, sine κ(g) 4 for every grph G onsiere in this setion. Before proving the neessity of Theorem 5, we hve the following ft. Ft 12 Let G e 4-onnete plnr grph with 4p verties where p > 4 n let (G) 5. Assume tht G hs extly two vertex isjoint fes F 1 n F 2 eh of whih hs extly p verties. If G is spnning sugrph of oughnut grph then G n e ugmente to 5-onnete 5-regulr grph G through tringultion of ll the non-tringulte fes of G exept the fes F 1 n F 2. One n esily oserve tht the following ft hols from the onstrution Construt-Doughnut given in Setion 3. Ft 13 Let G e oughnut grph, n let P e ny x-y pth suh tht V (P ) {V (F 1 ) V (F 2 )} = n x hs extly two neighors on fe F 1 (F 2 ). Then the following onitions (i) n (ii) hol. (i) If P is even, then the vertex y hs extly two neighors on fe F 1 (F 2 ) n extly one neighor on fe F 2 (F 1 ). (ii) If P is o, then the vertex y hs extly one neighor on fe F 1 (F 2 ) n extly two neighors on fe F 2 (F 1 ). We re rey to prove the neessity of Theorem 5. Proof for the Neessity of Theorem 5 Assume tht G is spnning sugrph of p-oughnut grph. Then y Theorem 1 G hs 4p verties. Clerly (G) 5 n G stisfies the onitions (), () n (), otherwise G woul not e spnning sugrph of oughnut grph. The neessity of onition (e) is ovious y Ft 13. Hene it is suffiient to prove the neessity of onition () only. () G oes not hve ny fe of two or less verties sine G is 4-onnete plnr grph. Then every fe of G hs three or more verties. We now show tht G hs no fe of more thn four verties. Assume for ontrition tht G hs fe f of q verties suh tht q > 4. Then f n e tringulte y ing q 3 extr eges. These extr eges inrese the egrees of q 2 verties, n the sum of the egrees will e inrese y 2(q 3). Using the pigeonhole priniple, one n esily oserve tht there is vertex mong the q(> 4) verties whose egree will e rise y t lest 2 fter tringultion of f. Then G woul hve vertex of egree six or more where G is grph otine fter tringultion of f. Hene we nnot ugment G to 5-regulr grph through tringultion of ll the non-tringulte fes of G other thn the fes F 1 n F 2. Therefore G nnot e spnning sugrph of oughnut
JGAA, 13(2) 153 177 (2009) 169 grph y Ft 12, ontrition. Hene eh fe f of G exept F 1 n F 2 hs either three or four verties. In the remining of this setion we give onstrutive proof for the suffiieny of Theorem 5. Assume tht G stisfies the onitions in Theorem 5. We hve the following lemm. Lemm 14 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Assume tht ll the fes of G exept F 1 n F 2 re tringulte. Then G is oughnut grph. Proof: To prove the lim, we hve to prove tht (i) G is 5-onnete, (ii) G hs two vertex isjoint fes eh of whih hs extly p, p > 4, verties, n ll the other fes of G hs extly three verties, n (iii) G hs the minimum numer of verties stisfying the properties (i) n (ii). (i) We first prove tht G is 5-regulr grph. Every fe of G exept F 1 n F 2 is tringle. Furthermore eh of F 1 n F 2 hs extly p, p > 4, verties. Then G hs 3(4p) 6 2(p 3) = 10p eges. Sine none of the verties of G hs egree more thn five n G hs extly 10p eges, eh vertex of G hs egree extly five. We next prove tht the verties of G lie on three vertex-isjoint yles C 1, C 2 n C 3 suh tht yles C 1, C 2, C 3 ontin extly p, 2p n p verties, respetively. We tke n emeing Γ of G suh tht F 1 is emee s the outer fe n F 2 is emee s n inner fe. We tke the ontour of fe F 1 s yle C 1 n ontour of fe F 2 s yle C 3. Then eh of C 1 n C 2 ontins extly p, p > 4, verties. Sine G stisfies onitions (), () n () in Theorem 5 n ll the fes of G exept F 1 n F 2 re tringulte, the rest 2p verties of G form yle in Γ. We tke this yle s C 2. G(C 2 ) ontins C 3 sine G stisfies onition () in Theorem 5. Clerly C 1, C 2 n C 3 re vertex-isjoint n yles C 1, C 2, C 3 ontin extly p, 2p n p verties, respetively. We finlly prove tht G is 5-onnete. Assume for ontrition tht G hs ut-set of less thn five verties. In suh se G woul hve vertex of egree less thn five, ontrition. (ii) The proof of this prt is ovious sine G hs two vertex isjoint fes eh of whih hs extly p verties n ll the other fes of G hs extly three verties. (iii) The numer of verties of G is 4p. Using Lemm 2, we n esily prove tht the minimum numer of verties require to onstrut grph G tht stisfies the properties (i) n (ii) is 4p. We thus ssume tht G hs non-tringulte fe f exept fes F 1 n F 2. By onition () in Theorem 5, f is qurngle fe. It is suffiient to show tht we n ugment the grph G to oughnut grph y tringulting eh of the qurngle fes of G. However, we nnot ugment G to oughnut grph y tringulting eh qurngle fe ritrrily. For exmple, the grph G in Figure 7() stisfies ll the onitions in Theorem 5 n it hs extly one qurngle fe f 1 (,,, ). If we tringulte f 1 y ing n
170 Krim n Rhmn Stright-Line Gri Drwings on Liner Are ege (, ) s illustrte in Figure 7(), the resulting grph G woul not e oughnut grph sine oughnut grph oes not hve n ege (, ) suh tht V (F 1 ) n V (F 2 ). But if we tringulte f 1 y ing n ege (, ) s illustrte in Figure 7(), the resulting grph G is oughnut grph. Hene every tringultion of qurngle fe is not lwys vli to ugment G to oughnut grph. We ll tringultion of qurngle fe f of G vli tringultion if the resulting grph G otine fter the tringultion of f oes not ontrit ny onition in Theorem 5. We ll vertex v on the ontour of qurngle fe f goo vertex if v is one of the en vertex of n ege whih is e for vli tringultion of f. F 1 F 1 F 1 F 2 F 2 F 2 () () () Figure 7: () f 1 (,,, ) is qurngle fe, () the tringultion of f 1 y ing the ege (, ) n () the tringultion of f 1 y ing the ege (, ). We ll qurngle fe f of G n α-fe if f ontins t lest one vertex from eh of the fes F 1 n F 2. Otherwise, we ll qurngle fe f of G β-fe. In Figure 8, f 1 (,,, ) is n α-fe wheres f 2 (p, q, r, s) is β-fe. F 1 F 2 p q r s Figure 8: f 1 (,,, ) is n α-fe n f 2 (p, q, r, s) is β-fe. In vli tringultion of n α-fe f of G no ege is e etween ny two verties x, y V (f) suh tht x V (F 1 ) n y V (F 2 ). Hene the following ft hols on n α-fe f. Ft 15 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Let f e n α-fe in G. Then f mits unique vli tringultion n the tringultion is otine y ing n ege etween two verties those re not on F 1 n F 2.
JGAA, 13(2) 153 177 (2009) 171 Fes f 1 (,,, ) n f 2 (p, q, r, s) in Figure 9() re two α-fes n Figure 9() illustrtes the vli tringultions of f 1 n f 2. Verties n of f 1 n verties q n s of f 2 re goo verties. F 1 F 1 r F 2 r F 2 q s p () q s p () Figure 9: () f 1 (,,, ) n f 2 (p, q, r, s) re two α-fes, n () vli tringultions of f 1 n f 2. We ll β-fe β 1 -fe if the fe ontins extly one vertex either from F 1 or from F 2. Otherwise we ll β-fe β 2 -fe. In Figure 10, f 1 (,,, ) is β 1 -fe wheres f 2 (p, q, r, s) is β 2 -fe. We ll vertex v on the ontour of β 1 -fe f mile vertex of f if the vertex is in the mile position mong the three onseutive verties other thn the vertex on F 1 or F 2. In Figure 10, vertex of f 1 n vertex r of f 2 re the mile verties of f 1 n f 2, respetively. F 1 F 2 q p r s Figure 10: f 1 (,,, ) is β 1 -fe n f 2 (p, q, r, s) is β 2 -fe. In vli tringultion of β 1 -fe f of G no ege is e etween ny two verties x, y V (f) suh tht x, y / V (F 1 ) V (F 2 ). Hene the following ft hols on β 1 -fe f. Ft 16 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Let f e β 1 -fe of G. Then f mits unique vli tringultion n the tringultion is otine y ing n ege etween the vertex on F 1 or F 2 n the mile vertex. Fes f 1 (,,, ) n f 2 (p, q, r, s) in Figure 11() re two β 1 -fes n Figure 11() illustrtes the vli tringultions of f 1 n f 2. Verties n of f 1 n verties p n r of f 2 re goo verties.
172 Krim n Rhmn Stright-Line Gri Drwings on Liner Are F 1 F 1 F 2 F2 q r p () s q r p () s Figure 11: () f 1 (,,, ) n f 2 (p, q, r, s) re two β 1 -fes, n () vli tringultions of f 1 n f 2. In vli tringultion of β 2 -fe f of G no ege is e etween ny two verties x, y V (f) where x V (F 1 )(V (F 2 )), y / {V (F 1 ) V (F 1 )} n G hs either (i) n even q-y pth P suh tht q hs extly two neighors on F 2 (F 1 ) n V (P ) {V (F 1 ) V (F 2 )} =, or (ii) n o q-y pth P suh tht q hs extly two neighors on F 1 (F 2 ) n V (P ) {V (F 1 ) V (F 2 )} =. Hene the following ft hols on β 2 -fe f. Ft 17 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Let f e β 2 -fe of G. Then f mits unique vli tringultion n the tringultion is otine y ing n ege etween vertex on fe F 1 or F 2 n vertex z / V (F 1 ) V (F 2 ). Fe f 1 (,,, ) in the grph in Figure 12() is β 2 -fe n the grph hs n even u- pth P suh tht u hs extly two neighors g n h on F 2, n V (P ) {V (F 1 ) V (F 2 )} =. Figure 12() illustrtes the vli tringultion of f 1. Verties n re the goo verties of f 1. Fe f 2 (l, m, n, o) in the grph in Figure 12() is β 2 -fe n the grph hs n o v-o pth P suh tht v hs extly two neighors s n t on F 1, n V (P ) {V (F 1 ) V (F 2 )} =. Figure 12() illustrtes the vli tringultion of f 2. Verties l n n re the goo verties of f 2. Before giving proof for the suffiieny of Theorem 5 we nee to prove the following Lemms 18 n 19. Lemm 18 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Then ny qurngle fe f of G mits unique vli tringultion suh tht fter tringultion (v) 5 hols for ny vertex v in the resulting grph. Proof: By Fts 15, 16 n 17, f mits unique vli tringultion. Sine vli tringultion inreses the egree of goo vertex y one, it is suffiient to show tht eh goo vertex of f hs egree less thn five in G. Assume for ontrition tht goo vertex v hs egree more thn four in G. Then one n oserve tht G woul violte onition in Theorem 5.
JGAA, 13(2) 153 177 (2009) 173 F1 F 2 l F1 n o F 2 m F1 F 2 l F1 n o F 2 m h h g s v g s v u u () t t () () () Figure 12: Illustrtion for vli tringultion of β 2 -fe; () β 2 fe f 1 (,,, ) in grph stisfying onition (i), () β 2 fe f 2 (l, m, n, o) in grph stisfying onition(ii), () the vli tringultion of f 1 n () the vli tringultion of f 2. Lemm 19 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Also ssume tht G hs qurngle fes. Then no two qurngle fes f 1 n f 2 hve ommon vertex whih is goo vertex for oth the fes f 1 n f 2. Proof: Assume tht u is ommon vertex etween two qurngle fes f 1 n f 2. If u is neither goo vertex of f 1 nor goo vertex of f 2, then we hve one. We thus ssume tht u is goo vertex of f 1 or f 2. Without loss of generlity, we ssume tht u is goo vertex of f 1. Then u is not goo vertex of f 2, otherwise u woul not e ommon vertex of f 1 n f 2, ontrition. Proof for the Suffiieny of Theorem 5 Assume tht the grph G stisfies ll the onitions in Theorem 5. If ll the fes of G exept F 1 n F 2 re tringulte, then G is oughnut grph y Lemm 14. Otherwise, we tringulte eh qurngle fe of G, using its vli tringultion. Let G e the resulting grph. Lemms 18 n 19 imply tht (v) 5 for eh vertex v in G. Then the grph G stisfies the onitions in Theorem 5, sine G stisfies the onitions in Theorem 5, G is otine from G using vli tringultions of qurngle fes n (v) 5 for eh vertex v in G. Hene G is oughnut grph y Lemm 14. Therefore G is spnning sugrph of oughnut grph. We now hve the following lemm. Lemm 20 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Then G n e ugmente to oughnut grph in liner time. Proof: We first eme G suh tht F 1 is emee s the outer fe n F 2 is emee s n inner fe. We then tringulte eh of the qurngle fes of G using its vli tringultion if G hs qurngle fes. Let G e the resulting grph. As shown in the suffiieny proof of Theorem 5, G is
174 Krim n Rhmn Stright-Line Gri Drwings on Liner Are oughnut grph. One n esily fin ll qurngle fes of G n perform their vli tringultions in liner time, hene G n e otine in liner time. In Theorem 5 we hve given neessry n suffiient onition for 4- onnete plnr grph to e spnning sugrph of oughnut grph. As esrie in the proof of Lemm 20, we hve provie liner-time lgorithm to ugment 4-onnete plnr grph G to oughnut grph if G stisfies the onitions in Theorem 5. We hve thus ientifie sulss of 4-onnete plnr grphs tht mits stright-line gri rwings with liner re s stte in the following theorem. Theorem 6 Let G e 4-onnete plnr grph stisfying the onitions in Theorem 5. Then G mits stright-line gri rwing on gri of re O(n). Furthermore, the rwing of G n e foun in liner time. Proof: Using the metho esrie in the proof of Lemm 20, we ugment G to oughnut grph G y ing ummy eges (if require) in liner time. By Theorem 4, G mits stright-line gri rwing on gri of re O(n). We finlly otin rwing of G from the rwing of G y eleting the ummy eges (if ny) from the rwing of G. By Lemm 20, G n e ugmente to oughnut grph in liner time n y Theorem 4, stright-line gri rwing of oughnut grph n e foun in liner time. Moreover, the ummy eges n lso e elete from the rwing of oughnut grph in liner time. Hene the rwing of G n e foun in liner time. 6 Conlusion In this pper we introue new lss of plnr grphs, lle oughnut grphs, whih is sulss of 5-onnete plnr grphs. A grph in this lss hs stright-line gri rwing on gri of liner re, n the rwing n e foun in liner time. We showe tht the outerplnrity of oughnut grph is 3. Thus we ientifie sulss of 3-outerplnr grphs tht mits stright-line gri rwing with liner re. One n esily oserve tht ny spnning sugrph of oughnut grph lso mits stright-line gri rwing with liner re. However, the reognition of spnning sugrph of oughnut grph seems to e non-trivil prolem. We estlishe neessry n suffiient onition for 4-onnete plnr grph G to e spnning sugrph of oughnut grph. We lso gve liner-time lgorithm to ugment 4-onnete plnr grph G to oughnut grph if G stisfies the neessry n suffiient onition. By introuing the neessry n suffiient onition, in ft, we hve ientifie sulss of 4-onnete plnr grphs tht mits stright-line gri rwings with liner re. Fining other nontrivil lsses of plnr grphs tht mit stright-line gri rwings on gris of liner re is lso left s n open prolem.
JGAA, 13(2) 153 177 (2009) 175 Aknowlegements We thnk the referees n the orresponing eitor for their useful omments whih improve the presenttion of the pper.
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