Non-convex Representations of Graphs

Transcription

1 Non-conex Representations of Graphs Giseppe Di Battista, Fabrizio Frati, and Marizio Patrignani Dip. di Informatica e Atomazione Roma Tre Uniersity Abstract. We sho that eery plane graph admits a planar straight-line draing in hich all faces ith more than three ertices are non-conex polygons. 1 Introdction In a straight-line planar draing of a graph each edge is dran as a segment and no to segments intersect. A conex draing is a straight-line planar draing in hich each face is a conex polygon. While eery plane graph admits a planar straight-line draing [6], not eery plane graph admits a conex draing. Ttte shoed that eery triconnected plane graph admits sch a draing ith its oter face dran as an arbitrary conex polygon [12]. Thomassen [11] characterized the graphs admitting a conex draing and Chiba et al. [3] presented a linear-time algorithm for prodcing sch draings. Conex draings can be efficiently constrcted in small area [4,2]. Hong and Nagamochi proed that for eery triconnected plane graph hose bondary is a star-shaped polygon a draing in hich eery internal face is a conex polygon can be obtained in linear time [8]. The same athors also inestigated draings here the oter face is a conex polygon and the internal faces are star-shaped [9,10]. As opposed to traditional conex graph draing, some research orks explored the properties of draings ith non-conexity reqirements [7,1]. A pointed draing is sch that each ertex is incident to an angle larger than π. In [7] it is shon that a planar graph admits a straight-line pointed draing iff it is minimally rigid or a sbgraph of a minimally rigid graph. Since there exist planar graphs that do not admit straight-line pointed draings, algorithms that constrct pointed draings ith tangent-continos biarcs, circlar arcs, or parabolic arcs are stdied in [1]. In this paper e address the problem of prodcing non-conex draings, i.e., draings here all faces ith more than three ertices are non-conex. This can be considered as the opposite of the classic problem of constrcting conex draings. Also, it can be seen as the dal of the problem of constrcting pointed draings, since faces, and not ertices, are constrained to hae an angle greater than π. We proe the folloing: Theorem 1. Eery plane graph admits a non-conex draing. In Sect. 3 e proe the preios theorem for biconnected graphs by means of a constrctie algorithm hose indctie approach is reminescent of Fary s constrction [6], althogh it applies to non-trianglated graphs and relies on a more complex case stdy. De to space restrictions, some proofs are omitted and can be fond in [5]. In Sect. 4 e discss ho to extend the reslt to general plane graphs. Work partially spported by MUR nder Project MAINSTREAM: Algoritmi per strttre informatie di grandi dimensioni e data streams. I.G. Tollis and M. Patrignani (Eds.): GD 2008, LNCS 5417, pp , c Springer-Verlag Berlin Heidelberg 2009

2 Non-conex Representations of Graphs Preliminaries AgraphG is simple if it has no mltiple edges and no self-loops. A planar draing of G determines a circlar ordering of the edges incident to each ertex. To draings of G are eqialent if they determine the same circlar ordering arond each ertex. A planar embedding is an eqialence class of planar draings. A planar draing partitions the plane into faces. The nbonded face is the oter face and is denoted by f(g).achord of f(g) is an edge connecting to non-adjacent ertices of f(g). A graph ith a planar embedding and an oter face is called a plane graph.anon-conex draing of a plane graph G is a planar straight-line draing of G in hich each internal (external) face ith more thanthree ertices has an angle greaterthan π (smaller than π). Consider a face f of G and to non-adjacent ertices and incident to f. The contraction of and inside f leads to a graph G in hich and are replaced by asingleertex, connected to all ertices and are connected to in G. If and are connected to the same ertex y, theng contains to edges (, y) (hence G is not simple), nless (, y) and (y, ) are incident to f (in this case there is only one edge (, y)). A contraction can also be performed on to adjacent ertices and, by remoing edge (, ) and contracting and inside the face created by the remoal. 3 Proof of Theorem 1 In this section e assme that graph G is biconnected. The folloing lemmata hold. Lemma 1. Sppose that G has a face f ith at least for incident ertices. Then there exist to ertices and incident to f, sch that edge (, ) can be added to G inside f, so that the reslting plane graph G is simple. Lemma 2. Let G be a plane graph ith a face f haing more than for incident ertices. Let G be the plane graph obtained by inserting inside f an edge e beteen to non-adjacent ertices of f. Sppose that a non-conex draing Γ of G exists. Then the draing Γ obtained by remoing e from Γ is a non-conex draing of G. In order to proe Theorem 1 for biconnected plane graphs, e proe the folloing: Theorem 2. Let G be a biconnected plane graph sch that all the faces of G hae three or for incident ertices. Let f(g) be the oter face of G. Iff(G) has three ertices, then, for eery triangle T in the plane, G admits a non-conex draing in hich f(g) is represented by T.Iff(G) has for ertices and no chord, then, for eery non-conex qadrilateral Q in the plane, G admits a non-conex draing in hich f(g) is represented by Q. The mapping of the ertices of f(g) to the ertices of T or Q is arbitrary, proided that the circlar ordering of the ertices of f(g) is respected. Theorem 2, together ith Lemmata 1 and 2, implies Theorem 1. Namely, let G be any biconnected plane graph. While G has any face f ith at least fie ertices, add, by Lemma 1, a dmmy edge inside f so that the agmented graph is still plane and simple. The obtained graph G has (at least) one face ith for incident ertices for each face of G ith more than three incident ertices. In order to apply Theorem 2, consider

3 392 G. Di Battista, F. Frati, and M. Patrignani 2 4 l3,5 l 1, l 5 1,2 l4, (a) (b) Fig. 1. (a) G 6. (b) Constrction of a non-conex draing of G 6. f(g ) and, if f(g ) has for incident ertices and has a chord, insert a dmmy edge inside f(g ) beteen the to ertices not incident to the chord, trning f(g ) into a trianglar face. Constrct a non-conex draing of G, as in the proof of Theorem 2. By Lemma 2, remoing the dmmy edges inserted leaes the draing non-conex. Before proing Theorem 2, e gie to more lemmata. Let G 5 be the plane graph haing a 4-cycle (, 2, 3, 4 ) as oter face and one internal ertex 5 connected to, 2, 3,and 4.LetG 6 be the plane graph haing a 4-cycle (, 2, 3, 4 ) as oter face and to connected internal ertices 5 and 6 ith 5 connected to and 2,and 6 connected to 3 and 4 (see Fig. 1.a). Lemma 3. For any non-conex qadrilateral Q in the plane, there exists a non-conex draing of G 5 sch that f(g 5 ) is represented by Q. Lemma 4. For any non-conex qadrilateral Q in the plane, there exists a non-conex draing of G 6 sch that f(g 6) is represented by Q (see Fig. 1.b). Proof of Theorem 2. Let G be a biconnected plane graph haing all faces of size three or for and sch that, if f(g) has size for, then it has no chord. The proof is by indction on the nmber of internal ertices of G. In the base case, either G has no internal ertex and the statement is triially tre, or G = G 5,orG = G 6. In the latter cases the statement follos by Lemmata 3 and 4, respectiely. Indctiely assme that the statement holds for any biconnected plane graph ith less than n internal ertices. Sppose that G has n internal ertices. Three are the cases. G has a separating 3-cycle C. Denote by G 1 the graph obtained by remoing from G all ertices internal to C. Denote by G 2 the sbgraph of G indced by the ertices internal to and on the border of C. Iff(G 1 ) has three (for) ertices, then consider any triangle T (resp. any non-conex qadrilateral Q) and constrct a non-conex draing Γ 1 of G 1 haing T (resp. Q) as oter face. Consider the triangle T representing C in Γ 1. Constrct a non-conex draing Γ 2 of G 2 haing T as oter face and insert Γ 2 inside Γ 1 by gling the to draings along the common face C represented by T in both draings. The reslting draing Γ is a non-conexdraingof G. G has no separating 3-cycle and G has a separating 4-cycle C. Denote by G 1 the graph obtained by remoing from G all ertices internal to C. Denote by G 2 the sbgraph of G indced by the ertices internal to and on the border of C. Notice that f(g 2 ) has no chords, otherise G 2 (and then G) old hae a separating triangle. If

4 Non-conex Representations of Graphs z (a) (b) (c) Fig. 2. (a) Both and 2 are incident to f(g).(b)both and 2 are internal ertices of G.(c) Vertex is incident to f(g) and ertex 2 is not. f(g 1 ) has three (for) ertices, then consider any triangle T (resp. any non-conex qadrilateral Q) and constrct a non-conex draing Γ 1 of G 1 haing T (resp. Q) as oter face. Consider the non-conex qadrilateral Q representing C in Γ 1. Constrct a non-conex draing Γ 2 of G 2 haing Q as oter face and insert Γ 2 inside Γ 1 by gling the to draings along the common face C represented by Q in both draings. The reslting draing Γ is a non-conexdraingof G. G has no separating 3-cycle and no separating 4-cycle. We are going to contract to ertices of G to obtain a graph ith n 1 ertices in order to apply indction. Consider any internal ertex of G. First, e sho that at least one of the folloing statements holds: (i) there exist to trianglar faces incident to and sharing an edge (, ), sch that contracting edge (, ) does not create chords of f(g); (ii) there exists a qadrilateral face f =(,,, 2 ), sch that either contracting and or contracting and 2 does not create chords of f(g); (iii) G is G 5 ;(i)g is G 6. First, sppose that to trianglar faces (,, ) and (,, 2 ) exist. If also is internal to G, then contracting (, ) does not create chords of f(g) and statement (i) holds. Otherise, assme that is incident to f(g). Then, statement (i) does not hold only if both the folloing are tre: (a) f(g) is a 4-cycle and (b) there exists an edge connecting and the ertex of f(g) not adjacent to. In fact, if (a) does not hold, then f(g) is a 3-cycle and no chord can be generated by contracting any to ertices. Also, if (b) does not hold, then contracting to does not create chords of f(g). Then, sppose that f(g) is a 4-cycle (, 1,, 2 ) and that edge (, ) exists. If at least one of and 2 is internal to G, then either (,,, 1 ) or (,,, 2 ) is a separating 4-cycle, contradicting the hypotheses. Hence, assme that both and 2 are incident to f(g). The for 3-cycles (,, ), (,, 2 ), (,, ),and(,, 2 ) do not hae internal ertices by hypothesis. Hence, G contains no internal ertex other than, G = G 5, and statement (iii) holds. If to trianglar faces incident to and sharing an edge do no exist, then there exists a face f =(,,, 2 ). Edge (, ) does not belong to G, otherise either (,, ) or (,, 2 ) old be a separating 3-cycle. Analogosly, edge (, 2 ) does not belong to G. If is internal to G, then contracting and in f does not create chords of f(g) and statement (ii) holds. Otherise, ith similar argments as aboe, statement (ii) does not hold for and only if both the folloing are tre: (a) f(g) is a 4-cycle and (b) there exists an edge connecting and the ertex of f(g) not adjacent to.

5 394 G. Di Battista, F. Frati, and M. Patrignani l l 2 l l 2 l l 2 (a) (b) (c) Fig. 3. (a) and (b): Indctie constrction of Γ, hen statement (i) holds. The light shaded region represents disk D. (a) and (c): Indctie constrction of Γ, hen statement (ii) holds. The dark shaded region represents face (,,, 2) in Γ. Then, sppose that f(g) is a 4-cycle and that edge (, ) exists. We distingish three cases: (1) If both and 2 are incident to f(g) (see Fig. 2.a), then G contains 3-cycles (,,) and ( 2,,). SinceG has no separating triangle, sch cycles are faces of G, contradicting the hypothesis that to trianglar faces incident to and sharing an edge do no exist. (2) If both and 2 are internal ertices of G (see Fig. 2.b), then contracting and 2 does not create chords of f(g) and statement (ii) holds. (3) If one ot of and 2,say,isincidenttof(G) and the other, say 2, is not (see Fig. 2.c), consider the forth ertex z of f(g). If edge ( 2,z) does not exist, ertices and 2 can be contracted ithot creating chords of f(g) and statement (ii) holds. If edge ( 2,z) exists, there exist 3-cycles (,,) and (, 2,z), and4-cycles (,,, 2 ) and (,, z, 2 ). Sch cycles contain no ertices in their interior, since G has no separating 3-cycle or 4-cycle. Hence, G = G 6 and statement (i) holds. We proe that hicheer of the statements holds, a non-conex draing of G can be constrcted. Sppose that statement (i) holds, ith G haing faces (,, 1 ) and (,, 2 ). Contract edge (, ) to a ertex. The reslting graph G is simple. If not, then there old exist a ertex 3 adjacent to both and in G, ith 3 1, 2. This old imply that (,, 3 ) is a separating triangle, contradicting the hypotheses. Indctiely constrct a non-conex draing Γ of G (see Fig. 3.a). No consider the point p here ertex is dran in Γ. There exists a small disk D (see Fig. 3.b) centered at p sch that moing p to any point inside D leaes Γ a non-conex draing. Consider the line l throgh p and orthogonal to the line l connecting and 2. Remoe and its incident edges. Insert ertices and on l, so that both are inside D. Connect and to their neighbors. It is easy to see that all faces that had an angle greater than π in Γ still hae an angle greater than π in Γ and that the draing is still planar. No, sppose that statement (ii) holds. More precisely, sppose that contracting and insideafacef =(,,, 2 ) does not create chords of f(g). Contract and inside f to a ertex. The reslting graph G is simple. If not, then there old exist a ertex 3 adjacent to both and in G, ith 3, 2. This old imply that either (,,, 3 ) or (,, 2, 3 ) is a separating 4-cycle, contradicting the hypotheses. Indctiely constrct a non-conex draing Γ of G (see Fig. 3.a). Pertrb the ertices so that they are in general position and that Γ is still a non-conex draing. Consider the point p here is dran in Γ and consider the line l throgh p and orthogonal to the line l connecting and 2. There exists a small disk D (see Fig. 3.c) centered at p sch that D does not intersect l and sch that moing p to any point inside

6 Non-conex Representations of Graphs 395 D leaes Γ non-conex. Remoe and its incident edges. Insert ertices and on l, so that both are inside D. Connect and to their neighbors. It is easy to see that all faces that had an angle greater than π in Γ still hae an angle greater than π in Γ and that the draing is still planar. Frther, face (,,, 2 ) is non-conex, as ell, since the angle incident to the one of and farther from l is greater than π. Finally, sppose that statement (iii) or (i) holds. We are in one of the base cases and the claim directly follos from Lemmata 3 or 4, respectiely. 4 Conclsions We hae proed that eery biconnected plane graph admits a draing in hich each face ith more than three ertices has an angle greater than π. This reslt can be extended to general plane graphs as follos. Any simply-connected graph G can be sitably agmented to a biconnected graph G by adding dmmy edges in sch a ay that for each face of G ith more than three ertices there exists a corresponding face of G ith more than three ertices. Remoing dmmy edges from a non-conex draing of G yields a non-conex draing of G. We beliee it is of interest to determine hether non-conex draings can be realized on a polynomial-size grid. References 1. Aichholzer, O., Rote, G., Schlz, A., Vogtenhber, B.: Pointed draings of planar graphs. In: Bose, P. (ed.) CCCG 2007, pp (2007) 2. Bárány, I., Rote, G.: Strictly conex draings of planar graphs. Doc. Math. 11, (2006) 3. Chiba, N., Yamanochi, T., Nishizeki, T.: Linear algorithms for conex draings of planar graphs. In: Bondy, J.A., Mrty, U.S.R. (eds.) Progress in Graph Theory, pp Academic Press, Ne York (1984) 4. Chrobak, M., Goodrich, M.T., Tamassia, R.: Conex draings of graphs in to and three dimensions. In: Symposim on Comptational Geometry, pp (1996) 5. Di Battista, G., Frati, F., Patrignani, M.: Non-conex representations of graphs. Tech. Report RT-DIA , Dip. Informatica e Atomazione, Uni. Roma Tre (2008) 6. Fary, I.: On straight line representations of planar graphs. Acta. Sci. Math. 11, (1948) 7. Haas, R., Orden, D., Rote, G., Santos, F., Seratis, B., Seratis, H., Soaine, D., Strein, I., Whiteley, W.: Planar minimally rigid graphs and psedo-trianglations. Compt. Geometry Theory Appl. 31, (2005) 8. Hong, S., Nagamochi, H.: Conex draings of graphs ith non-conex bondary. In: Fomin, F.V. (ed.) WG LNCS, ol. 4271, pp Springer, Heidelberg (2006) 9. Hong, S., Nagamochi, H.: Star-shaped draings of planar graphs. In: Brankoich, L., Lin, Y., Smyth, W.F. (eds.) IWOCA College Pblications (2007) 10. Hong, S., Nagamochi, H.: Star-shaped draing of planar graphs ith fixed embedding and concae corner constraints. In: H, X., Wang, J. (eds.) COCOON LNCS, ol. 5092, pp Springer, Heidelberg (2008) 11. Thomassen, C.: Plane representations of graphs. In: Progress in Graph Theory, pp Academic Press, London (1984) 12. Ttte, W.T.: Conex representations of graphs. Proc. London Math. Soc. 10, (1960)