Straight-line Drawings of Outerplanar Graphs in O(dn log n) Area

Transcription

1 CCCG 2007, Ottaa, Ontaio, August 20 22, 2007 Staight-ine Daings of Outepana Gaphs in O(dn og n) Aea Fabizio Fati Abstact We sho an agoithm fo constucting O(dn og n) aea outepana staight-ine daings of n-etex outepana gaphs ith degee d. Aso, e sette in the negatie a conjectue [1] on the aea equiement of outepana gaphs by shoing that snofake gaphs admit inea aea daings. 1 Intoduction Amost thity yeas ago, Leiseson [7] and Doe and Tickey [4] independenty shoed that eey n-etex outepana gaph hose degee is bounded by 4 admits a poyine daing in Θ(n) aea. The techniques pesented in [7, 4] can be modified in ode to obtain poy-ine daings of outepana gaphs ith degee d in O(d 2 n) aea, as pointed out in [1]. Moe ecenty, the pobem of obtaining minimum aea daings of outepana gaphs has been tacked by Bied, ho in [1] poided an O(n og n) aea uppe bound fo poy-ine daings of genea outepana gaphs. Moeoe, she conjectued that thee exists a cass of outepana gaphs caed snofake gaphs equiing Ω(n og n) aea in any pana staight-ine o poy-ine daing. Concening staight-ine daings, in [5] Gag and Rusu hae shon that eey n-etex outepana gaph has a staight-ine daing ith O(dn 1.48 ) aea. Hoee, a sub-quadatic aea bound has been poed to yeas ago in [3], hee Di Battista and Fati shoed that fo staight-ine daings of genea outepana gaphs O(n 1.48 ) aea aays suffices. In Section 3 e poide an agoithm fo obtaining staight-ine daings of outepana gaphs in O(dn og n) aea. Futhe, in Section 4 e sho that snofake gaphs admit inea aea daings, setting in the negatie the aboe cited conjectue appeaed in [1]. In the same section e gie concusions and a conjectue concening the aea equiement of staight-ine daings of outepana gaphs. 2 Peiminaies We assume famiiaity ith Gaph Daing. Fo basic definitions see aso [2]. A pana staight-ine gid daing of a gaph is such that each etex is mapped to a point ith intege coodinates, each edge is epesented by a segment beteen its endpoints Dipatimento di Infomatica e Automazione, Uniesità di Roma Te, fati@dia.unioma3.it Wok patiay suppoted by MUR unde Poject MAINSTREAM Agoithms fo Massie Infomation Stuctues and Data Steams. and no to edges coss, but, possiby, at common endpoints. In the fooing, uness otheise specified, fo daing e aays mean pana staight-ine gid daing. Ceay, a staight-ine daing of a gaph G is fuy detemined by the pacement of the etices of G. The aea of a daing is the numbe of gid points in the smaest ectange ith sides paae to the axes that coes the daing competey. An embedding of a gaph G is an odeing of the edges incident to each etex. An embedding of G detemines its dua gaph that is the gaph ith one etex pe face of G and ith one edge beteen to etices if the coesponding faces shae an edge in G. An outepana embedding is a pana embedding of a gaph in hich a etices ae incident to the same face, say the extena face. An outepana gaph is a gaph that admits an outepana embedding. The dua gaph of an outepana embedded gaph is a tee hen not consideing the etex coesponding to the extena face. A maxima outepana gaph is a gaph that admits an outepana embedding in hich a faces, but fo the extena one, ae tianges. The dua gaph of a maxima outepana embedded gaph is a binay tee. The degee of a etex is the numbe of edges incident to the etex. The degee of a gaph is the maximum degee of one of its etices. Let T be a binay tee ooted at node. Let T() denote the subtee of T ooted at node. The eftmost path L(T) (the ightmost path R(T)) of T is path 0, 1,..., k such that 0 =, i+1 is the eft chid (esp. the ight chid) of i, i such that 0 i k 1, and k doesn t hae a eft chid (esp. a ight chid). The eft-ight (ight-eft) path of a node T is path 0, 1,..., k such that 0 =, 1 is the eft (esp. ight) chid of 0, and 1, 2,..., k is the ightmost (esp. eftmost) path of T( 1 ). Conside any daing Γ of T. The eft poygon of the neighbos P () (the ight poygon of the neighbos P ()) of a node T is the poygon of the segments epesenting in Γ the edges of the eft-ight (esp. ight-eft) path pus a segment connecting k and 0. A pana staight-ine daing Γ of a binay tee T is stashaped if (1) it s ode-peseing, i.e. the ode of the chiden is the same of one fixed in adance; (2) fo each node T, P () = (, 1,..., k ) and P () = (, u 1,..., u p ) ae simpe poygons and each segment (, i ), ith 2 i k 1 (esp. (, u j ), ith 2 j p 1), beongs to the inteio of P () (esp. P ()), but fo its endpoints; (3) fo each pai of nodes a, b T, each of P ( a ) and P ( a ) doesn t intesect P ( b ) and P ( b ), but, possiby, at common endpoints o at common edges; and (4) thee exist points p and p fom hich it s possibe to da edges to each node of

2 19th Canadian Confeence on Computationa Geomety, 2007 L(T) and to each node of R(T), espectiey, and to da edge (p, p ), ithout ceating cossings ith the edges of Γ. Let E be an outepana embedding of a maxima outepana gaph G and et T be the dua binay tee of E. Seect any edge (u, u ) on the extena face of E and oot T at the intena face containing (u, u ). We ca etices u and u poes of G and e aso ca intena subgaph the gaph obtained by deeting u, u and thei incident edges fom G. In [3] the staight-ine daabiity of an n-etex outepana gaph G as sticty eated to the sta-shaped daabiity of its dua tee T, by means of the fooing emma: Lemma 1 [3] If T admits a sta-shaped daing ith f(n) aea, then G has an outepana staight-ine daing hee the aea of the daing of its intena subgaph is f(n). The poof of the peious emma is based on a bijection that can be estabished beteen the nodes of T and the etices of G, but fo its poes. Such a bijection aos to da G by augmenting the sta-shaped daing of T ith to exta points (epesenting the poes of G) and ith some exta staight-ine edges. Hence, in a sta-shaped daing of T ith aea f(n), it is sufficient to guaantee a pacement fo the poes of G not asymptoticay inceasing f(n) to be abe to constuct a daing of G in O(f(n)) aea. 3 The agoithm The outine of the agoithm is as foos: (i) augment the input outepana gaph G to a maxima outepana gaph G ; (ii) seect any edge (u, u ) on the extena face of the outepana embedding E of G and oot the dua binay tee T of E at the intena face of E containing (u, u ); constuct a sta-shaped daing Γ of T ; (iii) inset the poes of G and the edges that ae needed to augment Γ in a daing Γ of G ; (i) emoe the dummy edges inseted in the fist step to obtain a daing of G. Step (i) can be pefomed by means of the agoithm descibed in [6], hee it is shon that an outepana gaph can be augmented to maxima by inseting dummy edges that do not asymptoticay ate the degee of the gaph. No e descibe ho to constuct a sta-shaped daing Γ of T. The outine of such a constuction is as foos. A path S is emoed fom T, togethe ith the edges incident to the etices of S. The subtees that ae disconnected fom the emoa of S ae ecusiey dan. Path S is chosen so that each one of such subtees is sma, that is, has at most n/2 nodes. The daings of the ecusiey dan subtees ae hoizontay aigned, namey they ae a contained in an hoizonta stip H. The etica extension of such a stip is gien by the height of the highest daing of a subtee ecusiey dan. The nodes of S ae dan in the uppe pat and in the oe pat of the daing, that is, in the 4d+1 hoizonta gid ines aboe and beo H, espectiey. In paticua, S is patitioned in subpaths, and each subpath cuts H, that is, is dan in pat aboe and in pat beo H. The daing is constucted to satisfy the isibiity popeties of a sta-shaped daing. In paticua, the eftmost and the ightmost path of T (and of any subtee ecusiey dan) ae paced on the oest ine intesecting the daing. Denote by h 1 and h 2 the hoizonta gid ines deimiting H ith h 1 aboe h 2 at a etica distance that i be detemined ate. The 4d + 1 hoizonta gid ines aboe h 1 (esp. beo h 2 ), that compose the uppe pat (esp. the oe pat) of the daing ae abeed u 1, u 2,..., u 4d+1 (esp. 1, 2,..., 4d+1 ) fom the oest to the highest (esp. fom the highest to the oest). A nodes of L(T) and of R(T) ie on 4d+1. Assume T is ooted at any node of degee at most 2. Seect a path S = ( 0, 1,..., m ) in T, caed spine, such that (a) 0 =, (b) fo 1 i k, i is oot of the subtee ith the geatest numbe of nodes among the subtees ooted at chiden of i 1, and (c) m is a eaf. The nodes of T beonging (not beonging) to S ae spine nodes (non-spine nodes). We ca eft edge (ight edge) an edge ( i 1, i ) of S such that i is the eft (ight) chid of i 1 in S. Path S is subdiided into subpaths, based on the atenance beteen eft and ight edges that compose S. Iteatiey subdiide S in paths S 0, S 1,..., S q, so that, fo 0 j q, S j = ( j k, j k+1,...,j, j +1,..., j f 1, j f ) is defined as foos: k 0 = 0, and, fo 1 j q, j k is the node afte j 1 f in S. Suppose that ( j k, j k+1 ) is a ight edge. Let j be the fist node afte j k in S such that (j, j +1 ) is a eft edge. If ( j +1, j +2 ) is a eft edge (ight edge) then et j f be the fist node afte j in S such that (j f, j f+1 ) is a ight (esp. eft) edge. No suppose that ( j k, j k+1 ) is a eft edge. Let j be the fist node afte j k in S such that (j, j +1 ) is a ight edge. If ( j +1, j +2 ) is a ight edge (eft edge) then et j f be the fist node afte j in S such that (j f, j f+1 ) is a eft (esp. ight) edge. Notice that S q j coud hae no etex q o q f if the spine ends befoe a eft edge o a ight edge is encounteed. T is aso subdiided in subtees: Fo 0 j q, T j is the subtee of T induced by the nodes in S j and the nodes in the subtees ooted at non-spine nodes chiden of spine nodes in S j. No e sho ho to constuct a daing Γ j of each T j, fo 0 j q. We distinguish eight cases, based on hethe (i) j is een (Cases ) o odd (Cases ); (ii) ( j k, j k+1 ) is a ight edge (Cases ) o a eft edge (Cases ); and (iii) ( j +1, j +2 ) is a ight edge (Cases ) o a eft edge (Cases ). In the cases in hich j is een, da L(T j ) and R(T j ) on 2d+1 so that the each node of L(T j ) (of R(T j )) is one unit to the ight (to the eft) of its eft (ight) chid. In the cases in hich j is odd, da L(T j ) and R(T j ) on u 2d+1 so that the each node of L(T j ) (of R(T j )) is one unit to the eft (to the ight) of its eft (ight) chid. Denote by h the etica gid ine passing though, and denote by h +1, h +2, h 1, and h 2 the etica gid ines one unit to the ight, to units to the ight, one unit to

3 CCCG 2007, Ottaa, Ontaio, August 20 22, 2007 u 4d+1 u 4d Case 1 0 f-1 G 0 G 1 Case 5 G 2 f 1 Case u 2d+2 u 2d+1 u 1 h 1 H h 2 1 2d+1 2d+2 4d 4d+1 G 0 0 f 0 k 0 G 0 G 1 1 f 1 1 k f-1 G 1 G 2 G 2 G 2 2 f k G 2 Figue 1: Thick segments epesent the edges of S. Left (ight) edges ae abeed by (esp. by ). the eft, and to units to the eft of h, espectiey. Case 1. (see Fig. 1) Da j f at the intesection beteen h +1 and u 2d+2 ; da R(T( j f )) on h +1, ith any node one unit aboe its ight chid. Da j f 1 at the intesection beteen h 2 and u 4d+1. Da R(T( j +1 )) ti j f 1 on h 2, ith any node one unit beo its ight chid; da L(T( j +1 )) on h 2, ith any node one unit aboe its eft chid. Case 2. Da j f at the intesection beteen h 2 and u 4d+1 ; da L(T( j f )) on h 2, ith any node one unit aboe its eft chid. Da j f 1 at the intesection beteen h +1 and u 2d+2. Da L(T( j +1 )) ti j f 1 on h +1, ith any node one unit beo its eft chid; da R(T( j +1 )) on h +1, ith any node one unit aboe its ight chid. Case 3. Da j f at the intesection beteen h +2 and u 2d+2 ; da R(T( j f )) on h +2, ith any node one unit aboe its ight chid. Da j f 1 at the intesection beteen h 1 and u 4d+1. Da R(T( j +1 )) ti j f 1 on h 1, ith any node one unit beo its ight chid; da L(T( j +1 )) on h 1, ith any node one unit aboe its eft chid. Case 4. (see Fig. 1) Da j f at the intesection beteen h 1 and u 4d+1 ; da L(T( j f )) on h 1, ith any node one unit aboe its eft chid. Da j f 1 at the intesection beteen h +2 and u 2d+2. Da L(T( j +1 )) ti j f 1 on h +2, ith any node one unit beo its eft chid; da R(T( j +1 )) on h +2, ith any node one unit aboe its ight chid. Case 5. (see Fig. 1) Da j f at the intesection beteen h 1 and 4d+1 ; da R(T( j f )) on h 1, ith any node one unit beo its ight chid. Da j f 1 at the intesection beteen h +2 and 2d+2. Da R(T( j +1 )) on h +2, ith any node one unit aboe its ight chid; da L(T( j +1 )) on h +2, ith any node one unit beo its eft chid. Case 6. Da j f at the intesection beteen h +2 and 2d+2 ; da L(T( j f )) on h +2, ith any node one unit beo its eft chid. Da j f 1 at the intesection beteen h 1 and 4d+1. Da L(T( j +1 )) on h 1, ith any node one unit aboe its eft chid; da R(T( j +1 )) on h 1, ith any node one unit beo its ight chid. Case 7. Da j f at the intesection beteen h 2 and 4d+1 ; da R(T( j f )) on h 2, ith any node one unit beo its ight chid. Da j f 1 at the intesection beteen h +1 and u 2d+2. Da R(T( j +1 )) on h +1, ith any node one unit aboe its ight chid; da L(T( j +1 )) on h +1, ith any node one unit beo its eft chid. Case 8. Da j f at the intesection beteen h +1 and 2d+2 ; da L(T( j f )) on h +1, ith any node one unit beo its eft chid. Da j f 1 at the intesection beteen h 2 and 4d+1. Da L(T( j +1 )) on h 2, ith any node one unit aboe its eft chid; da R(T( j +1 )) on h 2, ith any node one unit beo its ight chid. In Γ 0 shift the nodes of L(T 0 ) and R(T 0 ) eticay, so that they keep the same x-coodinates and ie on ine 4d+1. Fo each T j, ith 0 j q, ecusiey constuct a daing of each subtee ooted at a node of T j that has not been aeady dan and that is chid of a node of T j that has been aeady dan. Let h max be the maximum beteen the heights of the daings of the subtees ecusiey dan. Set the distance beteen h 1 and h 2 to be h max 1, that is H consists of h max hoizonta gid ines. Fo each T j, ith 0 j q and j een, conside the daings of the subtees ooted at non-aeady dan nodes of T j that ae chiden of nodes beonging to L(T j ) o to R(T j ), and hose paents ae paced to the ight (to the eft) of j. Pace such daings in the eft-ight ode induced by the ode of thei paents on 4d+1 o on 2d+1, at one unit of hoizonta distance beteen them, ith the eftmost (esp. ightmost) etica ine intesecting the eftmost (esp. ightmost) daing one unit to the ight (esp. to the eft) of the ast node of R(T j ) (esp. of L(T j )), and ith thei eftmost and ightmost paths on h 1. Then, conside the daings of

4 19th Canadian Confeence on Computationa Geomety, 2007 u (a) u (b) 2 oot u 25 u 24 u u 25 u 24 u u 15 u 14 u 13 u u 15 u 14 u 13 u 12 u 2 u 1 h 1 =h u 2 u 1 h 1 =h (c) (d) Figue 2: (a) A maxima outepana gaph G ith n = 34 etices and degee d = 6. The poes of G ae abeed by u and u. (b) The dua binay tee T of G. Edges abeed () ae beteen a node and its eft (ight) chid. Thick edges sho the spine S seected by the agoithm descibed in Section 3. Red, geen, and bue etices sho subpaths S 0, S 1, and S 2 of S, espectiey. Cases 2, 8, and 1 hae to be appied to da S 0, S 1, and S 2, espectiey. (c) The sta shaped daing Γ of T constucted by the agoithm shon in Section 3. (d) The outepana daing of G obtained by augmenting Γ ith the poes of G and ith exta edges. the subtees ooted at non-aeady dan nodes of T j that ae chiden of nodes aeady dan. Rotate such daings of π adiants and pace them so that L(T j ) and R(T j ) ie on h 2, so that the daings of the subtees ooted at chiden of nodes dan on h +1 o on h +2 (on h 1 o on h 2 ) ae paced to the ight (esp. to the eft) of the daing constucted up to no, at one unit of hoizonta distance in the ode induced by thei paents in L(T( j +1 )) o in R(T(j +1 )). If j is odd, a daing Γ j of T j can be constucted anaogousy. No pace a the Γ j s togethe, stating fom Γ 0, and iteatiey adding Γ j, fo j = 1,...,m, so that the eftmost etica ine intesecting Γ j is one unit to the ight of the ightmost etica ine intesecting Γ j 1. The constucted daing Γ of T is sta-shaped (see the appendix). Let s anayze the aea equiement of Γ. The idth of Γ is tiiay O(n). The height of Γ is the sum of the heights of H, of the uppe pat, and of the oe pat of Γ. The height of H is equa to the height of the highest subtee of T ecusiey dan; by definition of S, each subtee ecusiey dan has at most n/2 nodes. Denoting by H(n)

5 CCCG 2007, Ottaa, Ontaio, August 20 22, 2007 the maximum height of a daing of an n-nodes tee T constucted by the agoithm, e get: H(n) = (4d+1)+(4d+ 1) + H(n/2) = O(d) + H(n/2) = O(d og n). Pace the poes of G both on the hoizonta ine one unit beo 0, so that they ae at a distance of one hoizonta unit, and so that one poe is on the same etica ine of 0. Notice that this pacement doesn t asymptoticay incease the aea of Γ. Finay, the edges necessay to augment Γ in a daing of G can be inseted and the dummy edges inseted in the fist step can be emoed obtaining a daing of G. Figue 2 shos an exampe of appication of the agoithm descibed in this section. Theoem 2 Any n-etex outepana gaph of degee d has a staight-ine outepana daing in O(dn og n) aea. Staightfoady, e obtain the fooing: Cooay 3 Any n-etex outepana gaph ith constant degee has a staight-ine outepana daing in O(n og n) aea. 4 Concusions and Conjectues In this pape e hae shon that O(dn og n) aea is aays achieabe fo staight-ine daings of outepana gaphs. In [1] Bied conjectued an Ω(n og n) oe bound on the aea equiement of staight-ine and poy-ine daings of outepana gaphs. Moe pecisey, she exhibited a cass of outepana gaphs, the snofake gaphs shon in Fig. 3.a, fo hich she caimed: staight-ine daing agoithm has been pesented in [3]. Such an agoithm constucts daings in hich the etices of the intena subgaph that ae connected to the poes ie on to haf-ines 1 and 2 ith a common endpoint and ith sopes π/4 and π/4. The entie daing of the intena subgaph ies in the edge ith ange π/2 deimited by 1 and 2. Hence, consideing thee copies of the daing of the intena subgaph of a compete outepana gaph constucted by the agoithm in [3], otating them of 0, π, and 3π/2, espectiey, pacing the daings togethe, and inseting the thee etices that ae poes fo the thee compete outepana gaphs, an O( n) O( n) aea staight-ine daing of a snofake gaph can be obtained (see Fig. 3.c). We oud ike to point up that a knon agoithms fo constucting staight-ine daings of genea outepana gaphs ty to minimize the extension of the daing in ony one coodinate dimension, hie aoing the othe dimension to be O(n). Hoee, e beiee that O(n og n) aea cannot be achieed by squeezing the daing in ony one dimension, and that hence a compaction in both dimensions (o a poof that one dimension is Ω(n)) shoud be pusued. Conjectue 1 Thee exist n-etex outepana gaphs that in any staight-ine daing in hich one dimension is O(n) equie ω(og n) in the othe dimension. We notice that thee exist outepana gaphs such that both the idth and the height of any gid daing ae Ω(og n) (e.g., outepana gaphs containing a compete binay tee as a subgaph). Acknoedgements u u u u Thanks to Giuseppe Di Battista fo usefu comments and suggestions. (a) (b) (c) Figue 3: (a) A snofake gaph. (b) A snofake gaph subdiided in thee compete outepana gaphs. (c) Daing snofake gaphs in inea aea. The shaded egions coespond to the thee copies of the daing of the intena subgaph of a compete outepana gaph constucted by the agoithm in [3]. Conjectue A [1] Any poy-ine daing of the snofake gaph has Ω(n og n) aea. It can be obseed that an n-etex snofake gaph is composed of thee identica O(n)-etex compete outepana gaphs (see Fig. 3.b), defined in [3] as the outepana gaphs hose dua gaphs ae compete binay tees. Fo compete outepana gaphs an O( n) O( n) aea Refeences [1] T. C. Bied. Daing oute-pana gaphs in O(n og n) aea. In Gaph Daing, pages 54 65, [2] G. Di Battista, P. Eades, R. Tamassia, and I. G. Tois. Gaph Daing. Pentice Ha, [3] G. Di Battista and F. Fati. Sma aea daings of outepana gaphs. In Gaph Daing, pages , [4] D. Doe and H. W. Tickey. On inea aea embedding of pana gaphs. Technica epot, Stanfod, USA, [5] A. Gag and A. Rusu. Aea-efficient daings of outepana gaphs. In Gaph Daing, pages , [6] G. Kant and H. L. Bodaende. Tianguating pana gaphs hie minimizing the maximum degee. Inf. Comput., 135(1):1 14, [7] C. E. Leiseson. Aea-efficient gaph ayouts (fo VLSI). In FOCS, pages , 1980.

6 19th Canadian Confeence on Computationa Geomety, 2007 Appendix: Poof of Sta-Shaped Visibiity We sho that, gien a ooted binay tee T, dua of a maxima outepana gaph G, the agoithm descibed in Section 3 constucts a sta-shaped daing Γ of T. Fist, obsee that the caim that a subtees ecusiey dan ae contained inside H hods, since the distance beteen h 1 and h 2 is set equa to the height of the highest subtee ecusiey dan. Futhe, a nodes diecty dan ae in the uppe and oe pat of Γ. Namey, e hae that: (i) by constuction such nodes ae nee paced aboe u 4d+1 o beo 4d+1 and (ii) if such nodes ae paced in H, then it s easy to deduce by the constuction that thee exists a eftmost o a ightmost path of a subtee of T hose ength is geate than d; hoee, this oud impy that thee exists a etex of G hose degee is geate than d, since a the nodes of a eftmost o ightmost path of a subtee of T ae neighbos of the same node of G (see aso [3]) Γ satisfies popety (i) of a sta-shaped daing by constuction. Concening popety (i), L(T) and R(T) ie on the bottommost ine 4d+1 intesecting Γ. Hence, pacing the poes u and u of G one unit beo 4d+1 aos to da edges fom u and u to the nodes of L(T) and R(T) ithout ceating cossings ith Γ. No e anayze popeties (ii) and (iii) of a sta-shaped daing. Fo eey node ying in the uppe o in the oe pat of Γ, a nodes of P () and of P () ae paced eithe on h 1, o on h 2, o in the uppe, o in the oe pat of Γ. This aos to eify that popeties (ii) and (iii) of a sta-shaped daing ae satisfied by Γ sepaatey fo each ecusie step of the agoithm. Conside a subtee T j of T, as defined in Section 3. Fo each node of T j diffeent fom j 1, j, j f 1, and j f, one beteen P () and P () has nodes paced eithe a on h 1 o a on h 2, but fo (see Fig. 4.a); the othe one beteen P () and P () has one node paced on the same hoizonta o etica ine of and the othe nodes eithe a on h 1 o a on h 2 (see Fig. 4.b). Hence, staight-ines can be dan fom to the nodes of P () and of P () ithout ceating cossings in Γ. Concening j 1 (j ) one beteen P ( j 1 ) and P ( j 1 ) (esp. one beteen P ( j ) and P ( j )) has nodes a on h 1 o a on h 2, but fo j 1 (esp. but fo j 1 and its chid that ies on the same hoizonta ine of j 1 ) and the othe one beteen P ( j 1 ) and P ( j 1 ) (esp. beteen P ( j ) and P ( j )) is a conex poygona ine (see Figs. 4.c and 4.d). Hence, staight-ines can be dan fom j 1 (fom j ) to the nodes of P ( j 1 ) and of P ( j 1 ) (esp. of P ( j ) and of P ( j )) ithout ceating cossings in Γ. Finay, conside nodes j f 1 and j f. One out of j f 1 and j f, say, is paced on u 4d+1 o on 4d+1, hie the othe one, say ˆ, is paced on u 2d+2 o on 2d+2. One beteen P () and P () has nodes a on h 1 o a on h 2, but fo, hie the othe one has nodes a on u 2d+1 o on 2d+1. To poe that is isibe fom the nodes of P () and P (), obsee that the sope of the edge connecting to j+1 k is ess than the one of the edge connecting to ˆ (see Fig. 4.e). Namey, the hoizonta (etica) distance beteen and j+1 k is at east 4 (is exacty 2d), hie the hoizonta (etica) distance beteen and ˆ is exacty 3 (is exacty 2d 1), and so the sope of (, j+1 k ) is ess o equa than 2d 2d 1 4, hie the one of (, ˆ) is 3. We hae that 2d 4 < 2d 1 3 if and ony if d > 2 that is aays satisfied consideing maxima outepana gaphs ith moe than 3 etices. Hence, staight-ines can be dan fom to the nodes of P () and of P () ithout ceating cossings in Γ. Concening ˆ, the nodes of one beteen P (ˆ) and P (ˆ) ie a on h 1, o a on h 2, but fo ˆ and eentuay fo its chid that ies on the same etica ine of ˆ. The nodes of the othe one beteen P (ˆ) and P (ˆ) ie a on u 2d+1 o a on 2d+1, but fo ˆ and eentuay fo, that e hae aeady poed to be isibe fom ˆ. Hence, staight-ines can be dan fom ˆ to the nodes of P (ˆ) and of P (ˆ) ithout ceating cossings in Γ (see Fig. 4.f). j j j -1 (a) (b) (c) (d) 3 4 (e) (f) j+1 k j+1 k u 4d+1 u 2d+2 u 2d+1 u 4d+1 u 2d+2 u 2d+1 Figue 4: Iustations fo the poof of sta-shaped isibiity of the daings constucted by the agoithm in Section 3.