A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without ﬁxed edges, determining which graphs always share a simultaneous embedding with ﬁxed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE

Fowler, J. Joseph

Jünger, Michael

Kobourov, Stephen G.

Schulz, Michael

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Characterizing Simultaneous Embedding with Fixed EdgesJ. Joseph Fowler1, Michael Ju¨nger2, Stephen G. Kobourov1, and Michael Schulz21 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the samevertex set V in the plane without crossings between edges of the same graph. Fixed edges are commonedges between graphs that share the same Jordan curve in the simultaneous drawings. While any numberof planar graphs have a simultaneous embedding without fixed edges, determining which graphs alwaysshare a simultaneous embedding with fixed edges (SEFE) has been open. We partially close this problemby giving a necessary condition to determine when pairs of graphs have a SEFE.1 IntroductionIn many practical applications including the visualization of large graphs and very-large-scale in-tegration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex setcan suffice in which multiple edge sets correspond to different edge colors or circuit layers. Whilethe union of any pair of edge sets may be nonplanar, a planar drawing of each layer may stillbe possible, as crossings between edges of distinct edge sets is permitted. This corresponds to theproblem of simultaneous embedding (SE).Without restrictions on the types of edges used, this problem is trivial since any number ofplanar graphs can be drawn on the same fixed set of vertex locations [6]. However, difficultiesarise once straight-line edges are required. Moving one vertex to reduce crossings in one layercan introduce additional crossings in other layers. This is the problem of simultaneous geometricembedding (SGE). If edge bends are allowed, then having common edges drawn the same way isimportant for mental map preservation. Such edges are called fixed edges leading to the problem ofsimultaneous embedding with fixed edges (SEFE). Since straight-line edges between a pair of verticesare also fixed, any graph that has a SGE also has a SEFE, but the converse is not true; see Fig. 1.Deciding if two graphs have a SGE is NP-hard [2], whereas, deciding if three graphs have a SEFEis NP-complete [4]. However, deciding if two graphs have a SEFE in polynomial-time remains open.In this paper we present a necessary condition in terms of forbidden subgraphs for whether pairs ofgraphs always have a SEFE. While this does not yet lead to a polynomial-time decision algorithmin the general case, it does in the more restricted case of pairs of outerplanar graphs. Additionally,we characterize which pairs of planar graphs and which pairs of outerplanar graphs have a SEFEand provide simultaneous drawing algorithms when possible.(b)(a) (c) (d)fff fih gcebdaih gebdcadcea bdcea bFig. 1. The path and planar graph in (a) do not have a SGE with straight-line edges [1], but but have a SEFE in (b).The two outerplanar graphs in (c) do not have a SEFE, but have a SE in (d) if edge (b, e) is not fixed.⋆ This work is supported in part by NSF grants CCF-0545743 and ACR-0222920.⋆⋆ This work is supported in part by the German Science Foundation (JU204/11-1) and by the German AcademicExchange Service.(a) (b) (c) (d) (e)Fig. 2. Forests in (a), circular caterpillars in (b), and subgraphs of K4 in (c) have a SEFE with any planar graph. K3-cyclesas in (d) and outerplanar graphs composed of cubic K3-cycles as in (e) have a SEFE with any outerplanar graph.1.1 Our ContributionWe omit the proofs of claims for the following results in this extended abstract, but details can befound in the technical report [3].1. While most pairs of graphs whose union forms a subdivided K5 or K3,3 share a SEFE, we give 16minimal forbidden pairs that do not. This gives a necessary condition for SEFE of two graphs.2. Using this condition we show that the only graphs that always have a SEFE with any planargraph are either (i) forests, (ii) circular caterpillars (removal of all degree-1 vertices leaves acycle), and (iii) subgraphs of K4; see Fig. 2(a)–(c). We also show that the the only outerplanargraphs that always share a SEFE with any outerplanar graph either are (i) biconnected in whichthe endpoints of every chord are at a distance of two from each along the outerface (K3-cycle)or (ii) have a cut vertex in which no two chords can be incident in the same biconnectedcomponent (each biconnected subgraph is a cubic K3-cycle); see Fig. 2(d)–(e). In each case, weprovide O(n2 lg n) simultaneous drawing algorithms.3. By comparing all possible subsets of vertices in a pair of outerplanar graphs against the twoforbidden outerplanar pairs, we have a necessary condition for SEFE. We show that this conditionalso suffices, giving a O(n2) time decision algorithm for SEFE of two outerplanar graphs.2 Forbidden Simultaneous Embeddings with Fixed EdgesWe start by stating the seminal theorem by Kuratowski [5] that characterizes all planar graphs.Theorem 1 (Kuratowski) Every nonplanar graph has a subgraph homeomorphic to K5 or K3,3.Suppose a pair of graphs G1(V,E1) and G2(V,E2) does not have a SEFE as in Fig. 3(a). Ifdeleting any edge from either graph allows a SEFE, then G1 and G2 are edge minimal as is Fig. 3(b).If a degree-2 vertex v (adjacent to u and w) in the union is not a degree-1 vertex in either G1 or G2,then we can unsubdivide the vertex by deleting v and replacing edges (u, v) and (v,w) with the edge(u,w) in G1 and/or G2. A pair of graphs for which this can no longer be done is vertex minimal asis Fig. 3(c). A minimal forbidden pair does not have a SEFE and is edge and vertex minimal. Analternating edge is a u  v path in which the edges strictly alternate between being in either G1and G2, but not both. An exclusive edge is a u v path composed of the single edge (u, v) only inG1 or G2, while an inclusive edge is composed of the single edge (u, v) in the intersection.Claim 2 Any pair of graphs G1(V,E1) and G2(V,E2) can be reduced to a pair in which every u vpath is either an inclusive edge, an exclusive edge or an alternating edge.Any pair of graphs for which this has been done for all u  v paths is called a reduced pair.Suppose G1 and G2 are a reduced pair. The alternating edge subgraph, denoted G1 ⊎ G2, is thesubgraph of G1∪G2 consisting only of alternating edges. The exclusive edge subgraph of G1, denotedG1\G2, is the subgraph of G1∪G2, of exclusive edges from G1, where G2\G1 is defined analogously.Hence, edges of G1∪G2 are partitioned into G1⊎G2, G1\G2, G2\G1, and G1∩G2; see Fig. 3(c)–(g).Next we see why we only need to consider crossings between nonincident edges.2(a) (b) (c) (d) (e) (f) (e)G1 ∪ G2 G1 ⊎ G2G1 ∩ G2 G1 \ G2 G2 \ G1Fig. 3. Removing extra edges from (a) gives (b). Unsubdividing vertices gives (c) with the four subgraphs (d)–(g).Observation 3 Crossings in a nonplanar drawing between a pair of incident edges can be removedwithout affecting the number of crossings of nonincident edges.Applying this observation to K5 and K3,3 of Theorem 1 gives the following corollary.Corollary 4 (a) Every drawing of K5 or K3,3 has at least one crossing between nonincident edges.(b) Any K5 and K3,3 can be drawn with only one crossing between any pair of nonincident edges.We use this corollary to produce a sufficient condition for SEFE.Lemma 5 Suppose the union of a reduced pair G1 and G2 is homeomorphic to K5 or K3,3. Letu v and x y be nonincident paths in G1 ∪G2 but not in G1 ∩G2. G1 and G2 share a SEFE ifeither path belongs to G1 ⊎G2 or one belongs to G1 \G2 and the other belongs to G2 \G1.Applying Lemma 5, we next determine when a pair of graphs forming K5 or K3,3 has a SEFE.Corollary 6 Suppose the union of a reduced pair G1 and G2 is homeomorphic to K5 or K3,3.G1 and G2 do not share a SEFE if and only if (i) every nonincident edge of an alternating edge inG1⊎G2 is in G1∩G2 and (ii) every nonincident edge of an exclusive edge in G1 \G2 is also in G1.Using these two conditions we are able to show the following theorem.ce a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbcace a eb d dbca(c)(a) (d)(b)(f) (g)(e)(h) (i) (j)G3,2G3,1G1,2G1,1 G4,2G4,1G2,2G2,1G6,2G6,1 G7,2G7,1G5,2G5,1G8,2G8,1 G9,2G9,1 G10,2G10,1Fig. 4. Ten K5 minimal forbidden pairs. The dark edges are in G1 ∩G2 and the dashed edges are in G1 ⊎ G2.a b cd ea b cd ef fa b cd ea b cd ef fa b cd ea b cd ef fa b cd ea b cd ef fa b cd ea b cd ef fa b cd ea b cd ef f(a) (b)(e)(d) (f)(c)G11,1 G11,2 G12,1 G12,2G15,1 G15,2G14,1 G14,2 G16,1 G16,2G13,1 G13,2Fig. 5. Six K3,3 minimal forbidden pairs. The dark edges are in G1 ∩G2 and the dashed edges are in G1 ⊎G2.3Theorem 7 There are 16 minimal forbidden pairs whose union is homeomorphic to K5 or K3,3.We next consider when edge contraction does not change when a pairs of graphs has a SEFE.Lemma 8 Let G1 and G2 be a pair of graphs with a SEFE. Let G′1 and G′2 be the graphs aftercontracting edge (u, v) in G1 ∩ G2 to vertex w. If there do not exist edges (u, x) in G1 \ G2 and(v, x) in G2 \G1 (or vice versa), then G′1 and G′2 have a SEFE.We extend our definition of minimality to include edge contraction to give the next lemma.Lemma 9 The only minimal forbidden pairs are the 16 pairs homeomorphic to K5 or K3,3.3 Characterizing Simultaneous Embeddings with Planar GraphsIn this section we determine which graphs always have a SEFE with any planar graph and showhow to produce a simultaneous drawing. Let P be the set of planar graphs and PSEFE ⊂ P be thesubset of forests, circular caterpillars and subgraphs of K4.Lemma 10 The set PSEFE contains the graphs that can always have a SEFE with any planar graph.Graphs in PSEFE not only can, but always have a SEFE with any planar graph as we show next.Theorem 11 Pair G1 ∈ PSEFE, G2 ∈ P on n vertices have a SEFE computable in O(n2 lg n) time.4 Characterizing Simultaneous Embeddings with Outerplanar GraphsWe next determine which outerplanar graphs always have a SEFE with any other outerplanargraph and produce simultaneous drawings when possible. Let O be the set of outerplanar graphsand OSEFE ⊂ O be the set of outerplanar graphs that consist of a single K3-cycle or have a cubicK3-cycle for each biconnected subgraph.Lemma 12 The set OSEFE contains the outerplanar graphs that can always have a SEFE with anyouterplanar graph.We next show outerplanar graphs in OSEFE not only can, but do indeed have a SEFE with anyouterplanar graph.Theorem 13 Pair G1 ∈ OSEFE, G2 ∈ O on n vertices have a SEFE computable in O(n2 lg n) time.An example layout is shown in Fig. 6 in the appendix. We conclude with our last theorem.Theorem 14 Deciding if a pair of outerplanar graphs have a SEFE can be done in O(n2) time.References1. P. Brass, E. Cenek, C. A. Duncan, A. Efrat, C. Erten, D. Ismailescu, S. G. Kobourov, A. Lubiw, and J. S. B.Mitchell. On simultaneous graph embedding. Computational Geometry: Theory and Applications, 36(2):117–130,2007.2. A. Estrella-Balderrama, E. Gassner, M. Ju¨nger, M. Percan, M. Schaefer, and M. Schulz. Simultaneous geometricgraph embeddings. In S. Hong and T. Nishizeki, editors, 15th Symposium on Graph Drawing, GD 2007, volume4875 of LNCS, pages 280–290. Springer, Heidelberg, 2008.3. J. J. Fowler, M. Ju¨nger, S. G. Kobourov, and M. Schulz. Characterizing simultaneous embedding with fixed edges.Technical Report TR08-01, University of Arizona, 2008. ftp://ftp.cs.arizona.edu/reports/2008/TR08-01.pdf.4. E. Gassner, M. Ju¨nger, M. Percan, M. Schaefer, and M. Schulz. Simultaneous graph embeddings with fixed edges.In F. V. Fomin, editor, 32nd International Workshop on Graph-Theoretic Concepts in Computer Science, WG2006, volume 4271 of LNCS, pages 325–335. Springer, Heidelberg, 2006.5. C. Kuratowski. Sur les proble`mes des courbes gauches en Topologie. Fundamenta Mathematicae, 15:271–283,1930.6. J. Pach and R. Wenger. Embedding planar graphs at fixed vertex locations. In Proceedings of the 6th Symposiumon Graph Drawing, pages 263–274, 1998.4Appendix(a) G1 (b) G2(c) Planar drawing of G1 with fixed edgess using vertex locations of G2.Fig. 6. SEFE of two outerplanar graphs. The edges of the four biconnected component of G1 are colored distinctly.5

Characterizing Simultaneous Embeddings with Fixed Edges

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