We show that every plane graph admits a planar straight-line drawing in which all faces with more than three vertices are non-convex polygon

C. Thomassen

I. Bárány

I. Fary

N. Chiba

R. Haas

S. Hong

W.T. Tutte

Di Battista, Giuseppe

Frati, Fabrizio

Patrignani, Maurizio

Graph Drawing E-print Archive

Non-Convex Representations of Graphs

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English

Non-convex Representations of Graphs

GDEA

DI BATTISTA, Giuseppe

FRATI, FABRIZIO

PATRIGNANI, Maurizio

Archivio della Ricerca - Università di Roma 3

Non-convex Representations of GraphsGiuseppe Di Battista, Fabrizio Frati, and Maurizio PatrignaniDip. di Informatica e Automazione – Roma Tre UniversityAbstract. We show that every plane graph admits a planar straight-line drawingin which all faces with more than three vertices are non-convex polygons.1 IntroductionIn a straight-line planar drawing of a graph each edge is drawn as a segment and notwo segments intersect. A convex drawing is a straight-line planar drawing in whicheach face is a convex polygon. While every plane graph admits a planar straight-linedrawing [6], not every plane graph admits a convex drawing. Tutte showed that ev-ery triconnected plane graph admits such a drawing with its outer face drawn as anarbitrary convex polygon [12]. Thomassen [11] characterized the graphs admitting aconvex drawing and Chiba et al. [3] presented a linear-time algorithm for producingsuch drawings. Convex drawings can be efficiently constructed in small area [4,2].Hong and Nagamochi proved that for every triconnected plane graph whose bound-ary is a star-shaped polygon a drawing in which every internal face is a convex polygoncan be obtained in linear time [8]. The same authors also investigated drawings wherethe outer face is a convex polygon and the internal faces are star-shaped [9,10].As opposed to traditional convex graph drawing, some research works explored theproperties of drawings with non-convexity requirements [7,1]. A pointed drawing issuch that each vertex is incident to an angle larger than π. In [7] it is shown that a planargraph admits a straight-line pointed drawing iff it is minimally rigid or a subgraph ofa minimally rigid graph. Since there exist planar graphs that do not admit straight-linepointed drawings, algorithms that construct pointed drawings with tangent-continuousbiarcs, circular arcs, or parabolic arcs are studied in [1].In this paper we address the problem of producing non-convex drawings, i.e., draw-ings where all faces with more than three vertices are non-convex. This can be consid-ered as the opposite of the classic problem of constructing convex drawings. Also, it canbe seen as the dual of the problem of constructing pointed drawings, since faces, andnot vertices, are constrained to have an angle greater than π. We prove the following:Theorem 1. Every plane graph admits a non-convex drawing.In Sect. 3 we prove the previous theorem for biconnected graphs by means of a con-structive algorithm whose inductive approach is reminescent of Fary’s construction [6],although it applies to non-triangulated graphs and relies on a more complex case study.Due to space restrictions, some proofs are omitted and can be found in [5]. In Sect. 4we discuss how to extend the result to general plane graphs. Work partially supported by MUR under Project “MAINSTREAM: Algoritmi per struttureinformative di grandi dimensioni e data streams.”I.G. Tollis and M. Patrignani (Eds.): GD 2008, LNCS 5417, pp. 390–395, 2009.c© Springer-Verlag Berlin Heidelberg 2009Non-convex Representations of Graphs 3912 PreliminariesA graph G is simple if it has no multiple edges and no self-loops. A planar drawing of Gdetermines a circular ordering of the edges incident to each vertex. Two drawings of Gare equivalent if they determine the same circular ordering around each vertex. A planarembedding is an equivalence class of planar drawings. A planar drawing partitions theplane into faces. The unbounded face is the outer face and is denoted by f(G). A chordof f(G) is an edge connecting two non-adjacent vertices of f(G). A graph with a planarembedding and an outer face is called a plane graph. A non-convex drawing of a planegraph G is a planar straight-line drawing of G in which each internal (external) facewith more than three vertices has an angle greater than π (smaller than π).Consider a face f of G and two non-adjacent vertices u and v incident to f . Thecontraction of u and v inside f leads to a graph G′ in which u and v are replaced bya single vertex w, connected to all vertices u and v are connected to in G. If u andv are connected to the same vertex y, then G′ contains two edges (w, y) (hence G′ isnot simple), unless (u, y) and (y, v) are incident to f (in this case there is only oneedge (w, y)). A contraction can also be performed on two adjacent vertices u and v, byremoving edge (u, v) and contracting u and v inside the face created by the removal.3 Proof of Theorem 1In this section we assume that graph G is biconnected. The following lemmata hold.Lemma 1. Suppose that G has a face f with at least four incident vertices. Then thereexist two vertices u and v incident to f , such that edge (u, v) can be added to G insidef , so that the resulting plane graph G′ is simple.Lemma 2. Let G be a plane graph with a face f having more than four incident ver-tices. Let G′ be the plane graph obtained by inserting inside f an edge e between twonon-adjacent vertices of f . Suppose that a non-convex drawing Γ ′ of G′ exists. Thenthe drawing Γ obtained by removing e from Γ ′ is a non-convex drawing of G.In order to prove Theorem 1 for biconnected plane graphs, we prove the following:Theorem 2. Let G be a biconnected plane graph such that all the faces of G havethree or four incident vertices. Let f(G) be the outer face of G. If f(G) has threevertices, then, for every triangle T in the plane, G admits a non-convex drawing inwhich f(G) is represented by T . If f(G) has four vertices and no chord, then, for everynon-convex quadrilateral Q in the plane, G admits a non-convex drawing in whichf(G) is represented by Q. The mapping of the vertices of f(G) to the vertices of T orQ is arbitrary, provided that the circular ordering of the vertices of f(G) is respected.Theorem 2, together with Lemmata 1 and 2, implies Theorem 1. Namely, let G be anybiconnected plane graph. While G has any face f with at least five vertices, add, byLemma 1, a dummy edge inside f so that the augmented graph is still plane and simple.The obtained graph G′ has (at least) one face with four incident vertices for each faceof G with more than three incident vertices. In order to apply Theorem 2, consider392 G. Di Battista, F. Frati, and M. Patrignaniu1 u2u5u3u4u6u1l1,3l3,5l1,2l4,5u2u5u3u4u6(a) (b)Fig. 1. (a) G∗6. (b) Construction of a non-convex drawing of G∗6.f(G′) and, if f(G′) has four incident vertices and has a chord, insert a dummy edgeinside f(G′) between the two vertices not incident to the chord, turning f(G′) into atriangular face. Construct a non-convex drawing of G′, as in the proof of Theorem 2.By Lemma 2, removing the dummy edges inserted leaves the drawing non-convex.Before proving Theorem 2, we give two more lemmata. Let G∗5 be the plane graphhaving a 4-cycle (u1, u2, u3, u4) as outer face and one internal vertex u5 connected tou1, u2, u3, and u4. Let G∗6 be the plane graph having a 4-cycle (u1, u2, u3, u4) as outerface and two connected internal vertices u5 and u6 with u5 connected to u1 and u2, andu6 connected to u3 and u4 (see Fig. 1.a).Lemma 3. For any non-convex quadrilateral Q in the plane, there exists a non-convexdrawing of G∗5 such that f(G∗5) is represented by Q.Lemma 4. For any non-convex quadrilateral Q in the plane, there exists a non-convexdrawing of G∗6 such that f(G∗6) is represented by Q (see Fig. 1.b).Proof of Theorem 2. Let G be a biconnected plane graph having all faces of size threeor four and such that, if f(G) has size four, then it has no chord.The proof is by induction on the number of internal vertices of G. In the base case, ei-ther G has no internal vertex and the statement is trivially true, or G = G∗5, or G = G∗6.In the latter cases the statement follows by Lemmata 3 and 4, respectively. Inductivelyassume that the statement holds for any biconnected plane graph with less than n inter-nal vertices. Suppose that G has n internal vertices. Three are the cases.G has a separating 3-cycle C. Denote by G1 the graph obtained by removing fromG all vertices internal to C. Denote by G2 the subgraph of G induced by the verticesinternal to and on the border of C. If f(G1) has three (four) vertices, then consider anytriangle T (resp. any non-convex quadrilateral Q) and construct a non-convex drawingΓ1 of G1 having T (resp. Q) as outer face. Consider the triangle T ′ representing C inΓ1. Construct a non-convex drawing Γ2 of G2 having T ′ as outer face and insert Γ2inside Γ1 by gluing the two drawings along the common face C represented by T ′ inboth drawings. The resulting drawing Γ is a non-convex drawing of G.G has no separating 3-cycle and G has a separating 4-cycle C. Denote by G1the graph obtained by removing from G all vertices internal to C. Denote by G2 thesubgraph of G induced by the vertices internal to and on the border of C. Notice thatf(G2) has no chords, otherwise G2 (and then G) would have a separating triangle. IfNon-convex Representations of Graphs 393uvwu1u2uvwu1 u2uvzwu1 u2(a) (b) (c)Fig. 2. (a) Both u1 and u2 are incident to f(G). (b) Both u1 and u2 are internal vertices of G. (c)Vertex u1 is incident to f(G) and vertex u2 is not.f(G1) has three (four) vertices, then consider any triangle T (resp. any non-convexquadrilateral Q) and construct a non-convex drawing Γ1 of G1 having T (resp. Q) asouter face. Consider the non-convex quadrilateral Q′ representing C in Γ1. Construct anon-convex drawing Γ2 of G2 having Q′ as outer face and insert Γ2 inside Γ1 by gluingthe two drawings along the common face C represented by Q′ in both drawings. Theresulting drawing Γ is a non-convex drawing of G.G has no separating 3-cycle and no separating 4-cycle. We are going to contracttwo vertices of G to obtain a graph with n − 1 vertices in order to apply induction.Consider any internal vertex v of G. First, we show that at least one of the follow-ing statements holds: (i) there exist two triangular faces incident to v and sharing anedge (v, u), such that contracting edge (v, u) does not create chords of f(G); (ii) thereexists a quadrilateral face f = (v, u1, u, u2), such that either contracting v and u orcontracting u1 and u2 does not create chords of f(G); (iii) G is G∗5; (iv) G is G∗6.First, suppose that two triangular faces (u, v, u1) and (u, v, u2) exist. If also u isinternal to G, then contracting (u, v) does not create chords of f(G) and statement (i)holds. Otherwise, assume that u is incident to f(G). Then, statement (i) does not holdonly if both the following are true: (a) f(G) is a 4-cycle and (b) there exists an edgeconnecting v and the vertex w of f(G) not adjacent to u. In fact, if (a) does not hold,then f(G) is a 3-cycle and no chord can be generated by contracting any two vertices.Also, if (b) does not hold, then contracting v to u does not create chords of f(G).Then, suppose that f(G) is a 4-cycle (u,w1, w, w2) and that edge (v, w) exists. Ifat least one of u1 and u2 is internal to G, then either (u, v, w,w1) or (u, v, w,w2) isa separating 4-cycle, contradicting the hypotheses. Hence, assume that both u1 and u2are incident to f(G). The four 3-cycles (u, v, u1), (u, v, u2), (w, v, u1), and (w, v, u2)do not have internal vertices by hypothesis. Hence, G contains no internal vertex otherthan v, G = G∗5, and statement (iii) holds.If two triangular faces incident to v and sharing an edge do no exist, then there existsa face f = (v, u1, u, u2). Edge (u, v) does not belong to G, otherwise either (v, u, u1)or (v, u, u2) would be a separating 3-cycle. Analogously, edge (u1, u2) does not belongto G. If u is internal to G, then contracting v and u in f does not create chords off(G) and statement (ii) holds. Otherwise, with similar arguments as above, statement(ii) does not hold for v and u only if both the following are true: (a) f(G) is a 4-cycleand (b) there exists an edge connecting v and the vertex w of f(G) not adjacent to u.394 G. Di Battista, F. Frati, and M. Patrignaniu1ll’u2wu1ll’u2vuu1ll’u2vu(a) (b) (c)Fig. 3. (a) and (b): Inductive construction of Γ , when statement (i) holds. The light shaded regionrepresents disk D. (a) and (c): Inductive construction of Γ , when statement (ii) holds. The darkshaded region represents face (u, u1, v, u2) in Γ .Then, suppose that f(G) is a 4-cycle and that edge (v, w) exists. We distinguish threecases: (1) If both u1 and u2 are incident to f(G) (see Fig. 2.a), then G contains 3-cycles(u1, v, w) and (u2, v, w). Since G has no separating triangle, such cycles are facesof G, contradicting the hypothesis that two triangular faces incident to v and sharingan edge do no exist. (2) If both u1 and u2 are internal vertices of G (see Fig. 2.b),then contracting u1 and u2 does not create chords of f(G) and statement (ii) holds.(3) If one out of u1 and u2, say u1, is incident to f(G) and the other, say u2, is not(see Fig. 2.c), consider the fourth vertex z of f(G). If edge (u2, z) does not exist,vertices u1 and u2 can be contracted without creating chords of f(G) and statement (ii)holds. If edge (u2, z) exists, there exist 3-cycles (u1, v, w) and (u, u2, z), and 4-cycles(v, u1, u, u2) and (v, w, z, u2). Such cycles contain no vertices in their interior, since Ghas no separating 3-cycle or 4-cycle. Hence, G = G∗6 and statement (iv) holds.We prove that whichever of the statements holds, a non-convex drawing of G canbe constructed. Suppose that statement (i) holds, with G having faces (v, u, w1) and(v, u, w2). Contract edge (u, v) to a vertex w. The resulting graph G′ is simple. If not,then there would exist a vertex w3 adjacent to both u and v in G, with w3 = w1, w2.This would imply that (u, v, w3) is a separating triangle, contradicting the hypotheses.Inductively construct a non-convex drawing Γ ′ of G′ (see Fig. 3.a). Now considerthe point p where vertex w is drawn in Γ ′. There exists a small disk D (see Fig. 3.b)centered at p such that moving p to any point inside D leaves Γ ′ a non-convex drawing.Consider the line l′ through p and orthogonal to the line l connecting u1 and u2. Removew and its incident edges. Insert vertices u and v on l′, so that both are inside D. Connectu and v to their neighbors. It is easy to see that all faces that had an angle greater thanπ in Γ ′ still have an angle greater than π in Γ and that the drawing is still planar.Now, suppose that statement (ii) holds. More precisely, suppose that contracting uand v inside a face f = (u, u1, v, u2) does not create chords of f(G). Contract u and vinside f to a vertex w. The resulting graph G′ is simple. If not, then there would exist avertex w3 adjacent to both u and v in G, with w3 = u1, u2. This would imply that either(u, v, u1, w3) or (u, v, u2, w3) is a separating 4-cycle, contradicting the hypotheses.Inductively construct a non-convex drawing Γ ′ of G′ (see Fig. 3.a). Perturb the ver-tices so that they are in general position and that Γ ′ is still a non-convex drawing.Consider the point p where w is drawn in Γ ′ and consider the line l′ through p andorthogonal to the line l connecting u1 and u2. There exists a small disk D (see Fig. 3.c)centered at p such that D does not intersect l and such that moving p to any point insideNon-convex Representations of Graphs 395D leaves Γ ′ non-convex. Remove w and its incident edges. Insert vertices u and v onl′, so that both are inside D. Connect u and v to their neighbors. It is easy to see that allfaces that had an angle greater than π in Γ ′ still have an angle greater than π in Γ andthat the drawing is still planar. Further, face (u, v, u1, u2) is non-convex, as well, sincethe angle incident to the one of u and v farther from l is greater than π.Finally, suppose that statement (iii) or (iv) holds. We are in one of the base cases andthe claim directly follows from Lemmata 3 or 4, respectively. 4 ConclusionsWe have proved that every biconnected plane graph admits a drawing in which each facewith more than three vertices has an angle greater than π. This result can be extendedto general plane graphs as follows. Any simply-connected graph G can be suitablyaugmented to a biconnected graph G′ by adding dummy edges in such a way that foreach face of G with more than three vertices there exists a corresponding face of G′with more than three vertices. Removing dummy edges from a non-convex drawing ofG′ yields a non-convex drawing of G. We believe it is of interest to determine whethernon-convex drawings can be realized on a polynomial-size grid.References1. Aichholzer, O., Rote, G., Schulz, A., Vogtenhuber, B.: Pointed drawings of planar graphs.In: Bose, P. (ed.) CCCG 2007, pp. 237–240 (2007)2. Ba´ra´ny, I., Rote, G.: Strictly convex drawings of planar graphs. Doc. Math. 11, 369–391(2006)3. Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear algorithms for convex drawings of planargraphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 153–173. Aca-demic Press, New York (1984)4. Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and threedimensions. In: Symposium on Computational Geometry, pp. 319–328 (1996)5. Di Battista, G., Frati, F., Patrignani, M.: Non-convex representations of graphs. Tech. ReportRT-DIA-134-2008, Dip. Informatica e Automazione, Univ. Roma Tre (2008)6. Fary, I.: On straight line representations of planar graphs. Acta. Sci. Math. 11, 229–233(1948)7. Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D., Streinu,I., Whiteley, W.: Planar minimally rigid graphs and pseudo-triangulations. Comput. Geome-try Theory Appl. 31, 31–61 (2005)8. Hong, S., Nagamochi, H.: Convex drawings of graphs with non-convex boundary. In: Fomin,F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 113–124. Springer, Heidelberg (2006)9. Hong, S., Nagamochi, H.: Star-shaped drawings of planar graphs. In: Brankovich, L., Lin,Y., Smyth, W.F. (eds.) IWOCA 2007. College Publications (2007)10. Hong, S., Nagamochi, H.: Star-shaped drawing of planar graphs with fixed embedding andconcave corner constraints. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092,pp. 405–414. Springer, Heidelberg (2008)11. Thomassen, C.: Plane representations of graphs. In: Progress in Graph Theory, pp. 43–69.Academic Press, London (1984)12. 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Non-Convex Representations of Graphs

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