Intersection Graphs of L-Shapes and Segments in the Plane

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1 Intersection Grphs of -Shpes nd Segments in the Plne Stefn Felsner 1, Kolj Knuer 2, George B. Mertzios 3, nd Torsten Ueckerdt 4 1 Institut für Mthemtik, Technische Universität Berlin, Germny. 2 IRMM, Université Montpellier 2, Frnce. 3 School of Engineering nd Computing Sciences, Durhm University, UK. 4 Deprtment of Mthemtics, Krlsruhe Institute of Technology, Germny. felsner@mth.tu-berlin.de, kolj.knuer@mth.univ-montp2.fr, george.mertzios@durhm.c.uk, torsten.ueckerdt@kit.edu Abstrct. An -shpe is the union of horizontl nd verticl segment with common endpoint. These come in four rottions:,, nd. A k-bend pth is simple pth in the plne, whose direction chnges k times from horizontl to verticl. If grph dmits n intersection representtion in which every vertex is represented by n, n or, k-bend pth, or segment, then this grph is clled n {}-grph, {, }-grph, B k -VPG-grph or SEG-grph, respectively. Motivted by theorem of Middendorf nd Pfeiffer [Discrete Mthemtics, 108(1): , 1992], stting tht every {, }-grph is SEG-grph, we investigte severl known subclsses of SEG-grphs nd show tht they re {}-grphs, or B k -VPG-grphs for some smll constnt k. We show tht ll plnr 3-trees, ll line grphs of plnr grphs, nd ll full subdivisions of plnr grphs re {}-grphs. Furthermore we show tht ll complements of plnr grphs re B 19-VPG-grphs nd ll complements of full subdivisions re B 2-VPG-grphs. Here full subdivision is grph in which ech edge is subdivided t lest once. Keywords: intersection grphs, segment grphs, co-plnr grphs, k- bend VPG-grphs, plnr 3-trees. 1 Introduction nd Motivtion A segment intersection grph, SEG-grph for short, is grph tht cn be represented s follows. Vertices correspond to stright-line segments in the plne nd two vertices re djcent if nd only if the corresponding segments intersect. Such representtions re clled SEG-representtions nd, for convenience, the clss of ll SEG-grphs is denoted by SEG. SEG-grphs re n importnt subject of study strongly motivted from n lgorithmic point of view. Indeed, hving n intersection representtion of grph (in pplictions grphs often come This work ws prtilly supported by (i) the DFG ESF EuroGIGA projects COM- POSE nd GrDR, (ii) the EPSRC Grnt EP/K022660/1 nd (iii) the ANR Project EGOS: ANR-12-JS

2 long with such given representtion) my llow for designing better or fster lgorithms for optimiztion problems tht re hrd for generl grphs. More thn 20 yers go, Middendorf nd Pfeiffer [23], considered intersection grphs of xis-ligned -shpes in the plne, where n xis-ligned -shpe is the union of horizontl nd verticl segment whose intersection is n endpoint of both. In prticulr, -shpes come in four possible rottions:,,, nd. For subset X of these four rottions, e.g., X = {} or X = {, }, we cll grph n X-grph if it dmits n X-representtion, i.e., vertices cn be represented by -shpes from X in the plne, ech with rottion from X, such tht two vertices re djcent if nd only if the corresponding -shpes intersect. Similrly to SEG, we denote the clss of ll X-grphs by X. The question if n intersection representtion with polygonl pths or pseudo-segments cn be stretched into SEG-representtion is clssicl topic in combintoril geometry nd Oriented Mtroid Theory. Middendorf nd Pfeiffer prove the following interesting reltion between intersection grphs of segments nd -shpes. Theorem 1 (Middendorf nd Pfeiffer [23]). Every {, }-representtion hs combintorilly equivlent SEG-representtion. This theorem is best-possible in the sense tht there re exmples of {, }- grphs which re no SEG-grphs [6, 23], i.e., such {, }-representtions cnnot be stretched. We feel tht Theorem 1, which of course implies tht {, } SEG, did not receive lot of ttention in the ctive field of SEG-grphs. In prticulr, one could use Theorem 1 to prove tht certin grph clss G is contined in SEG by showing tht G is contined in {, }. For exmple, very recently Pwlik et l. [24] discovered clss of tringle-free SEG-grphs with rbitrrily high chromtic number, disproving fmous conjecture of Erdős [17], nd it is in fct esier to see tht these grphs re {}-grphs thn to see tht they re SEG-grphs. To the best of our knowledge, the stronger result G {, } hs never been shown for ny non-trivil grph clss G. In this pper we initite this reserch direction. We consider severl grph clsses which re known to be contined in SEG nd show tht they re ctully contined in {}, which is proper subclss of {, }. Whenever grph is not known (or known not) to be n intersection grph of segments or xis-ligned -shpes, one often considers nturl generliztions of these intersection representtions. Asinowski et l. [3] introduced intersection grphs of xis-ligned k-bend pths in the plne, clled B k -VPG-grphs. An (xis-ligned) k-bend pth is simple pth in the plne, whose direction chnges k times from horizontl to verticl. Clerly, B 1 -VPG-grphs re precisely intersection grphs of ll four -shpes; the union of B k -VPG-grphs for ll k 0 is exctly the clss STRING of intersection grphs of simple curves in the plne. Now if grph G / SEG is B k -VPG-grph for some smll k, then one might sy tht G is not fr from being SEG-grph. 2

3 1.1 Our Results nd Relted Work et us denote the clss of ll plnr grphs by PANAR. A recent celebrted result of Chlopin nd Gonçlves [5] sttes tht PANAR SEG, which ws conjectured by Scheinermn [25] in However, their proof is rther involved nd there is not much control over the kind of SEG-representtions. Here we give n esy proof for non-trivil subclss of plnr grphs, nmely plnr 3-trees. A 3-tree is n edge-mximl grph of treewidth 3. Every 3-tree cn be built up strting from the clique K 4 nd dding new vertices, one t time, whose neighborhood is the so-fr constructed grph is tringle. Theorem 2. Every plnr 3-tree is n {}-grph. It remins open to generlize Theorem 2 to plnr grphs of treewidth 3 (i.e., subgrphs of plnr 3-trees). On the other hnd it is esy to see tht grphs of treewidth t most 2 re {}-grphs. Chplick nd the lst uthor show in [8] tht plnr grphs re B 2 -VPG-grphs, improving on n erlier result of Asinowski et l. [3]. In [8] it is lso conjectured tht PANAR {}, which with Theorem 1 would imply the min result of [5], i.e., PANAR SEG. Considering line grphs of plnr grphs, one esily sees tht these grphs re SEG-grphs. Indeed, stright-line drwing of plnr grph G cn be interpreted s SEG-representtion of the line grph (G) of G. We prove the following strengthening result. Theorem 3. The line grph of every plnr grph is n {}-grph. Krtochvíl nd Kuběn [20] consider the clss of ll complements of plnr (co-plnr) grphs, CO-PANAR for short. They show tht CO-PANAR re intersection grphs of convex sets in the plne, nd sk whether CO-PANAR SEG. As the Independent Set Problem in plnr grphs is known to be NP-complete [14], Mx Clique is NP-complete for ny grph clss G CO-PANAR, e.g., intersection grphs of convex sets. Indeed, the longstnding open question whether Mx Clique is NP-complete for SEG [21] hs recently been nswered ffirmtively by Cbello, Crdinl nd ngermn [4] by showing tht every plnr grph hs n even subdivision whose complement is SEG-grph. The subdivision is essentil in the proof of [4], s it still remins n open problem whether CO-PANAR SEG [20]. The lrgest subclss of CO-PANAR known to be in SEG is the clss of complements of prtil 2- trees [13]. Here we show tht ll co-plnr grphs re not fr from being SEGgrphs. Theorem 4. Every co-plnr grph is B 19 -VPG grph. Theorem 4 implies tht Mx Clique is NP-complete for B k -VPG-grphs with k 19. On the other hnd, the Mx Clique problem for B 0 -VPG-grphs cn be solved in polynomil time, while Vertex Colorbility remins NPcomplete but llows for 2-pproximtion [3]. Middendorf nd Pfeiffer [23] show tht the complement of ny even subdivision of ny grph, i.e., every edge is 3

4 subdivided with non-zero even number of vertices, is n {, }-grph. This implies tht Mx Clique is NP-complete even for {, }-grphs. We consider full subdivisions of grphs, tht is, subdivision H of grph G where ech edge of G is subdivided t lest once. It is not hrd to see tht full subdivision H of G is in STRING if nd only if G is plnr, nd tht if G is plnr, then H is ctully SEG-grph. Here we show tht this cn be further strengthened, nmely tht H is in n {}-grph. Moreover, we consider the complement of full subdivision H of n rbitrry grph G, which is in STRING but not necessrily in SEG. Here, similr to the result of Middendorf nd Pfeiffer [23] on even subdivisions we show tht such grph H is not fr from being SEG-grph. Theorem 5. et H be full subdivision of grph G. (i) If G is plnr, then H is n {}-grph. (ii) If G is ny grph, then the complement of H is B 2 -VPG-grph. The grph clsses considered in this pper re illustrted in Figure 1. We shll prove Theorems 2, 3, 4 nd 5 in Sections 2, 3, 4 nd 5, respectively, nd conclude with some open questions in Section 6. line grphs of plnr grphs plnr 3-trees full subdivisions of plnr grphs SEG complements of even subdivisions plnr grphs B 1 B 2 B 19 COCOMP STRING complements of plnr grphs complements of full subdivisions Fig. 1. Grph clsses considered in this pper. 1.2 Relted Representtions In the context of contct representtions, where distinct segments or k-bend pths my not shre interior points, it is known tht every contct SEGrepresenttion hs combintorilly equivlent contct B 1 -VPG-representtion, 4

5 but not vice vers [19]. Contct SEG-grphs re exctly plnr mn grphs nd their subgrphs [10], which includes for exmple ll tringle-free plnr grphs. Very recently, contct {}-grphs hve been chrcterized [7]. Necessry nd sufficient conditions for stretchbility of contct system of pseudo-segments re known [1, 11]. et us lso mention the closely relted concept of edge-intersection grphs of pths in grid (EPG-grphs) introduced by Golumbic et l. [15]. There re some notble differences, strting from the fct tht every grph is n EPG-grph [15]. Nevertheless, nlogous questions to the ones posed bout VPG-representtions of STRING-grphs re posed bout EPG-representtions of generl grphs. In prticulr, there is strong interest in finding representtions using pths with few bends, see [18] for recent ccount. 2 Proof of Theorem 2 Proof. et G be plne 3-tree with fixed plne embedding. We construct n {}-representtion of G stisfying the dditionl property tht for every inner tringulr fce {, b, c} of G there exists subset of the plne, clled the privte region of the fce, tht intersects only the -pths for, b nd c, nd no other -pth. More precisely, privte region of {, b, c} is n xis-ligned rectiliner polygon hving one of the shpes depicted in Figure 2(), such tht the -pths for, b nd c intersect the polygon s shown in figure. b c c b c b () (b) Fig. 2. () The two possible shpes of privte region for inner fcil tringle {, b, c}. (b) An {}-representtion of the plne 3-tree on three vertices together with privte region for the only inner fce. Indeed, we prove the following stronger sttement by induction on the number of vertices in G. Clim. Every plne 3-tree dmits n {}-representtion together with privte region for every inner fce, such tht the privte regions for distinct fces re disjoint. 5

6 As induction bse ( V (G) = 3) consider the grph G consisting only of tringle, b, c. Then there is n essentilly unique {}-representtion of G nd it is not difficult to find privte region for the unique inner fce of G. We refer to Figure 2(b) for n illustrtion. Now let us ssume tht V (G) 4. Becuse G is 3-tree there exists n inner vertex v of degree exctly three. In prticulr, the three neighbors, b, c of v form n inner fcil tringle in the plne 3-tree G = G \ v. By induction G dmits n {}-representtion with privte region for ech inner fce so tht distinct privte regions re disjoint. Consider the privte region R for {, b, c}. By flipping the plne long the min digonl if necessry, we cn ssume without loss of generlity tht R hs the shpe shown in the left of Figure 2(). (Note tht such flip does not chnge the type of the -pths.) Now we introduce n -pth for vertex v completely inside R s depicted in Figure 3(). Since R does not intersect ny other -pth this is n {}-representtion of G. b c b c v () v (b) Fig. 3. () Introducing n -shpe for vertex v into the privte region for the tringle {, b, c}. (b) Identifying pirwise disjoint privte regions for the fcil tringles {, b, v}, {, c, v} nd {b, c, v}. Finlly we identify three privte regions for the three newly creted inner fces {, b, v}, {, c, v} nd {b, c, v}. This is shown in Figure 3(b). Since these regions re pirwise disjoint nd completely contined in the privte region for {, b, c} we hve identified privte region for every inner fce so tht distinct regions re disjoint. (Note tht {, b, c} is not fcil tringle in G nd hence does not need privte region.) This proves the clim nd thus conclude the proof of the theorem. 3 Proof of Theorem 3 Proof. Without loss of generlity let G be mximlly plnr grph with fixed plne embedding. (ine grphs of subgrphs of G re induced subgrphs of (G).) Then G dmits so-clled cnonicl ordering, nmely n ordering v 1,..., v n of the vertices of G such tht Vertices v 1, v 2, v n form the outer tringle of G in clockwise order. (We drw G such tht v 1, v 2 re the highest vertices.) 6

7 For i = 3,..., n vertex v i lies in the outer fce of the induced embedded subgrph G i 1 = G[v 1,..., v i 1 ]. Moreover, the neighbors of v i in G i 1 form pth on the outer fce of G i 1 with t lest two vertices. We shll construct n {}-representtion of (G) long fixed cnonicl ordering v 1,..., v n of G. For every i = 2,..., n we shll construct n {}-representtion of (G i ) with the following dditionl properties. For every outer vertex v of G i there is bottomless rectngle R(v), i.e., n xis-ligned rectngle with bottom-edge t, such tht: R(v) intersects the horizontl segments of precisely those pths for edges in G i incident to v. R(v) does not contin ny bends or endpoints of ny pth for n edge in G i nd does not intersect ny R(w) for w v. the left-to-right order of the bottomless rectngles mtches the order of vertices on the counterclockwise outer v 1, v 2 -pth of G i. For i = 2, the grph G i consist only of the edge v 1 v 2. Hence n {}-representtion of the one-vertex grph (G 2 ) consists of only one -shpe nd two disjoint bottomless rectngles R(v 1 ), R(v 2 ) intersecting its horizontl segment. For i 3, let (w 1,..., w k ) be the counterclockwise outer pth of G i 1 tht corresponds to the neighbors of v i in G i 1. The corresponding bottomless rectngles R(w 1 ),..., R(w k ) pper in this left-to-right order. See Figure 4 for n illustrtion. For every edge v i w j, j = 1,..., k we define n -shpe P (v i w j ) whose verticl segment is contined in the interior of R(w j ) nd whose horizontl segment ends in the interior of R(w k ). Moreover, the upper end nd lower end of the verticl segment of P (v i w j ) lies on the top side of R(w j ) nd below ll -shpes for edges in G i 1, respectively. Finlly, the bend nd right end of P (v i w j ) is plced bove the bend of P (v i w j+1 ) nd to the right of the right end of P (v i w j+1 ) for j = 1,..., k 1, see Figure 4. v 1 v 2 R(w 1 ) R(w 2 ) R(w 3 ) R(w 4 ) w 2 G i 1 w 1 w 3 w4 v i R(v 1 ) R(v 2 ) R (w 1 ) R (v i ) R (w 4 ) Fig. 4. Along cnonicl ordering vertex v i is dded to G i 1. For ech edge between v i nd vertex in G i 1 n -shpe is introduced with its verticl segment in the corresponding bottomless rectngle. The three new bottomless rectngles R (w 1), R (v i), R (w k ) re highlighted. 7

8 It is strightforwrd to check tht this wy we obtin n {}-representtion of (G i ). So it remins to find set of bottomless rectngles, one for ech outer vertex of G i, stisfying our dditionl property. We set R (v) = R(v) for every v V (G i ) \ {v i, w 1,..., w k }. We define bottomless rectngle R (w 1 ) R(w 1 ) such tht R (w 1 ) is crossed by ll horizontl segments tht cross R(w 1 ) nd dditionlly the horizontl segment of P (v i w 1 ). Similrly, we define R (w k ) R(w k ). And finlly, we define R (V i ) R(w k ) in such wy tht it is crossed by the horizontl segments of exctly P (v i w 1 ),..., P (v i w k ). Note tht for 1 < j < k the outer vertex w j of G i 1 is not n outer vertex of G i. Then {R (v) v v(g i )} hs the desired property. See gin Figure 4. 4 Proof of Theorem 4 Proof. et G = (V, E) be ny plnr grph. We shll construct B k -VPG representtion of the complement Ḡ of G for some constnt k tht is independent of G. Indeed, k = 19 is enough. To find the VPG representtion we mke use of two crucil properties of G: A) G is 4-colorble nd B) G is 5-degenerte. Indeed, our construction gives B 2d+9 -VPG representtion for the complement of ny 4-colorble d-degenerte grph. Consider ny 4-coloring of G with color clsses V 1, V 2, V 3, V 4. Further let σ = (v 1,..., v n ) be n order of the vertices of V witnessing the degenercy of G, i.e., for ech v i there t most 5 neighbors v j of v i with nd j < i. We cll these neighbors the bck neighbors of v i. Consider ny ordered pir of color clsses, sy (V 1, V 2 ), nd denote W = V 1 V 2, together with the vertex orders inherited from the order of vertices in V, i.e., σ V1 = σ 1 = (v 1,..., v V1 ) nd σ V2 = σ 2 = (w 1,..., w V2 ). For v V 1 we denote the number of bck neighbors of v in V 2 by deg 2(v). Further consider the xis-ligned rectngle R = [0, A] [0, A], where A = 2( W +2). For illustrtion we divide R into four qurters [0, A/2] [0, A/2], [0, A/2] [A/2, A], [A/2, A] [0, A/2] nd [A/2, A] [A/2, A]. We define monotone incresing pth Q(v) for ech v W s follows. See Figure 5 for n illustrtion. For v V 1 let i 1 < < i k be the indices of bck neighbors of v in V 2 nd i = mx{j {0} [σ 1 (v)] v j V 2 }, tht is, i is the lrgest index of vertex in V 2 tht comes before v in σ or i = 0 if there is no such vertex. Then we define the pth Q(v) so tht it strts t (1, 0), uses the horizontl lines t y = 2i j 1 for j = 1,..., k, y = 2i + 1 nd y = A 2σ 1 (v) in tht order, uses the verticl lines t x = 1, x = 2i j + 1 for j = 1,..., k nd x = A 2σ 1 (v) in tht order, nd finlly ends t (A, A 2σ 1 (v)). Note tht Q(v) voids the top-left qurter of R, hs exctly one bend t (A 2σ 1 (v), A 2σ 1 (v)) in the top-right qurter, goes bove the point (2i, 2i) in the bottom-left qurter if nd only if i i 1,..., i k nd i i. For w i V 2 the pth P (w i ) is defined nlogous fter rotting the rectngle R by 180 degrees nd swpping the roles of V 1 nd V 2. 8

9 R Fig. 5. The induced subgrph G[W ] for two color clsses W = V 1 V 2 of plnr grph G nd VPG representtion of its complement Ḡ[W ] in the rectngle [0, 2( W + 2)] [0, 2( W + 2)]. It is strightforwrd to check tht {Q(v) v W } is VPG representtion of Ḡ[W ] completely contined in R, where ech Q(v) strts nd ends t the boundry of R nd hs t most 3 + 2k bends, where k is the number of bck neighbors of v in W. Now we hve defined for ech pir of color clsses V i V j VPGrepresenttion of Ḡ[V i V j ]. For every vertex v V we hve defined three Q- pths, one for ech colors clss tht v is not in. In totl the three Q-pths for the sme vertex v hve t most 9 + 2k 19 bends, where k 5 is the bck degree of v. It remins to plce the six representtions of Ḡ[V i V j ] non-overlpping nd to connect the three Q-pths for ech vertex in such wy tht connections for vertices of different color do not intersect. This cn esily be done with two extr bends per pths, bsiclly becuse K 4 is plnr (we refer to Figure 6 for one wy to do this). Finlly, note tht the first nd lst segment of every pth in the representtion cn be omitted, yielding the climed bound. 5 Proof of Theorem 5 Proof. et G be ny grph nd H be subdivision of G in which ech edge is subdivided t lest once. Without loss of generlity we my ssume tht every edge of G is subdivided exctly once or twice. (i) Assuming tht G is plnr, we shll find n {}-representtion of H s follows. Assume without loss of generlity tht G is mximlly plnr. We 9

10 V 2 V 3 V 4 V 1 Fig. 6. Interconnecting the VPG representtions of Ḡ[Vi Vj] by dding t most two bends for ech vertex. The set of pths corresponding to color clss V i is indicted by single pth lbeled V i, i = 1, 2, 3, 4. consider br visibility representtion of G, i.e., vertices of G re disjoint horizontl segments in the plne nd edges re disjoint verticl segments in the plne whose endpoints re contined in the two corresponding vertex segments nd which re disjoint from ll other vertex segments. Such representtion for plnr tringultion exists e.g. by [22]. See Figure 7 for n illustrtion Fig. 7. A plnr grph G on the left, br visibility representtion of G in the center, nd n {}-representtion of full division of G on the right. Here, the edges {1, 2}, {1, 3} nd {3, 6} re subdivided twice. It is now esy to interpret every segment s n, nd replce n segment corresponding to edge tht is subdivided twice by two -shpes. et us simply refer to Figure 7 gin. (ii) Now ssume tht G = (V, E) is ny grph. We shll construct B 2 - VPG representtion of the complement H of H = (V W, E ) with monotone incresing pths only. First, we represent the clique H[V ]. et V = {v 1,..., v n } nd define for i = 1,..., n the 2-bend pth P (v i ) for vertex v i to strt t (i, 0), hve bends t (i, i) nd (i + n, i), nd end t (i + n, n + 1). See Figure 8 for n illustrtion. For convenience, let us cll these pths v-pths. 10

11 i + n 1 + n j + n 2n i + n 1 + n j + n 2n P (w ij ) P (w j ) P (v j ) P (v j ) P (v i ) P (v i ) P (w i ) 1 i j n 1 i j n Fig. 8. eft: Inserting the pth P (w ij) for single vertex w ij subdividing the edge v iv j in G. Right: Inserting the pths P (w i) nd P (w j) for two vertices w i, w j subdividing the edge v iv j in G. Next, we define for every edge of G the 2-bend pths for the one or two corresponding subdivision vertices in H. We cll these pths w-pths. So let v i v j be ny edge of G with i < j. We distinguish two cses. Cse 1. The edge v i v j is subdivided by only one vertex w ij in H. We define the w-pth P (w ij ) to strt t (j 1 4, i ), hve bends t (j 1 4, j ) nd (i + n 1 4, j ), nd end t (i + n 1 4, n + 1), see the left of Figure 8. Cse 2. The edge v i v j is subdivided by two vertices w i, w j with v i w i, v j w j E(H). We define the strt, bends nd end of the w-pth P (w i ) to be (j 1 4, i ), (j 1 4, j 1 4 ), (i + n 1 4, j 1 4 ) nd (i + n 1 4, n + 1), respectively. The strt, bends nd end of the w-pth P (w j ) re (j 1 2, i 1 4 ), (j 1 2, j ), (i + n 1 2, j ) nd (i + n 1 2, n + 1), respectively. See the right of Figure 8. It is esy to see tht every w-pth P (w) intersects every v-pth, except for the one or two v-pths corresponding to the neighbors of w in H. Moreover, the two w-pths in Cse 2 re disjoint. It remins to check tht the w-pths for distinct edges of G mutully intersect. To this end, note tht every w- pth for edge v i v j strts ner (j, i), bends ner (j, j) nd (i + n, j) nd ends ner (i+n, n). Consider two w-pths P nd P tht strt t (j, i) nd (j, i ), respectively, nd bend ner (j, j) nd (j, j ), respectively. If j = j then it is esy to check tht P nd P intersect ner (j, j). Otherwise, let j > j. Now if j > i, then P nd P intersect ner (j, i), nd if j i, then P nd P intersect ner (i + n, j ). Hence we hve found B 2 -VPG-representtion of H, s desired. et us remrk, tht in this representtion some w-pths intersect non-trivilly long some horizontl or verticl lines. However, this cn be omitted by slight nd pproprite perturbtion of endpoints nd bends of w-pths. 11

12 6 Conclusions nd Open Problems Motivted by Middendorf nd Pfeiffer s theorem (Theorem 1 in [23]) tht every {, }-representtion cn be stretched into SEG-representtion, we considered the question which subclsses of SEG-grphs re ctully {, }-grphs, or even {}-grphs. We proved tht this is indeed the cse for severl grph clsses relted to plnr grphs. We feel tht the question whether PANAR {, }, s lredy conjectured [8], is of prticulr importnce. After ll, this, together with Theorem 1, would give new proof for the fct tht PANAR SEG. Open Problem 1 Ech of the following is open. (i) When cn B 1 -VPG-representtion be stretched into combintorilly equivlent SEG-representtion? (ii) Is {, } = SEG B 1 -VPG? (iii) Is every plnr grph n {}-grph, or B 1 -VPG-grph? (iv) Does every plnr grph dmit n even subdivision whose complement is n {}-grph, or B 1 -VPG-grph? (v) Recognizing B k -VPG grphs is known to be NP-complete for ech k 0 [6]. Wht is the complexity of recognizing {}-grphs, or {, }-grphs? Sometimes it is of prticulr interest to find SEG-representtions using only few different slopes for the segments. While biprtite nd tringle-free plnr grphs re (contct) SEG-grphs using only two slopes [12] nd three slopes [9], respectively, open conjectures of Scheinermn [25] nd West [27] stte, tht 3- colorble nd generl plnr grphs hve SEG-representtions with only 3 slopes nd 4 slopes, respectively. Cn Middendorf nd Pfeiffer s theorem be used to obtin SEG-representtions with few slopes? Recll tht k 0 B k -VPG = STRING [3]. Chplick et l. [6] prove tht B k - VPG B k+1 -VPG for ll k 0 nd lso tht SEG B k -VPG for ech k 0, even if SEG is restricted to three slopes only. Another nturl subclss of STRING, which is in no inclusion-reltion with SEG, is the clss COCOMP of co-comprbility grphs [16]. However, one cn prove result similr to the previous one concerning B k -VPG-grphs nd STRING-grphs: There is no k N such tht B k -VPG COCOMP. A proof cn be given long the degrees of freedoms pproch of Alon nd Scheinermn [2], i.e., by counting the grphs in the respective sets. First, Alon nd Scheinermn consider the number P (n, t) of t-dimensionl posets on n elements. They show tht for fixed t the growth of the logrithm of this number behves like nt log n. et CC(n, t) be the number of cocomprbility grphs of t-dimensionl posets on n elements. With esy dpttions of the Alon nd Scheinermn proof we obtin log CC(n, t) n(t 1 o(1)) log n. Every pth of B k -VPG-representtion cn be encoded by k + 3 numbers. The question whether two pths intersect cn be nswered by looking t the signs of few low degree polynomils in 2k + 6 vribles evluted t the encodings of the two pths. This mens tht the clss B k -VPGhs k + 3 degrees of freedom. 12

13 Alon nd Scheinermn show how to use Wrren s Theorem [26] to get n upper bound on the size of such clss. The logrithm of the number of B k -VPGgrphs on n vertices is O(1)nk log n. Compring the numbers we find tht there re cocomprbility grphs of (k+ 2)-dimensionl posets tht hve no B k -VPG-representtion. On the other hnd it is esy to see, tht the co-comprbility grph of d-dimensionl poset is B d 1 - VPG-grph. Is there similr prmeter ensuring few bends in representtions of SEG-grphs? We know tht the number of slopes in the SEG-representtion is not the right nswer here. References 1. N. Aerts nd S. Felsner. Stright line tringle representtions. In Proceedings of the 21st Interntionl Symposium on Grph Drwing (GD), pges , N. Alon nd E. Scheinermn. Degrees of freedom versus dimension for continment orders. Order, 5:11 16, A. Asinowski, E. Cohen, M. C. Golumbic, V. imouzy, M. ipshteyn, nd M. Stern. Vertex intersection grphs of pths on grid. J. Grph Algorithms Appl., 16(2): , S. Cbello, J. Crdinl, nd S. ngermn. The clique problem in ry intersection grphs. Discrete & Computtionl Geometry, 50(3): , J. Chlopin nd D. Gonçlves. Every plnr grph is the intersection grph of segments in the plne: extended bstrct. In Proceedings of the 41st nnul ACM symposium on Theory of computing, STOC 09, pges , S. Chplick, V. Jelínek, J. Krtochvíl, nd T. Vyskočil. Bend-bounded pth intersection grphs: Susges, noodles, nd wffles on grill. In Proceedings of the 38th Grph-Theoretic Concepts in Computer Science (WG), pges , S. Chplick, S. G. Kobourov, nd T. Ueckerdt. Equilterl -contct grphs. In Proceedings of the 39th Grph-Theoretic Concepts in Computer Science (WG), pges , S. Chplick nd T. Ueckerdt. Plnr grphs s VPG-grphs. Journl of Grph Algorithms nd Applictions, 17(4): , N. de Cstro, F. J. Cobos, J. C. Dn, A. Márquez, nd M. Noy. Tringle-free plnr grphs nd segment intersection grphs. J. Grph Algorithms Appl., 6(1):7 26, H. de Frysseix nd P. O. de Mendez. Representtions by contct nd intersection of segments. Algorithmic, 47(4): , H. de Frysseix nd P. O. de Mendez. Stretching of Jordn rc contct systems. Discrete Applied Mthemtics, 155(9): , H. de Frysseix, P. O. de Mendez, nd J. Pch. A left-first serch lgorithm for plnr grphs. Discrete Comput. Geom., 13(3-4): , M. C. Frncis, J. Krtochvíl, nd T. Vyskočil. Segment representtion of subclss of co-plnr grphs. Discrete Mthemtics, 312(10): , M. R. Grey nd D. S. Johnson. Computers nd Intrctbility: A guide to the theory of NP-completeness. W. H. Freemn, M. C. Golumbic, M. ipshteyn, nd M. Stern. Edge intersection grphs of single bend pths on grid. Networks, 54(3): , M. C. Golumbic, D. Rotem, nd J. Urruti. Comprbility grphs nd intersection grphs. Discrete Mthemtics, 43(1):37 46,

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