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23
Simultaneous Geometric Graph Embeddings
"... Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is plana ..."
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Cited by 15 (5 self)
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Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NPhard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. 1
Simultaneous graph embeddings with fixed edges
 In 32nd Workshop on GraphTheoretic Concepts in Computer Science (WG
, 2006
"... Foundation (JU204/101). Abstract. We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that t ..."
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Cited by 13 (7 self)
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Foundation (JU204/101). Abstract. We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NPcomplete even for three planar graphs. For two planar graphs the complexity status is still open. 1
Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges
, 2008
"... Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determinin ..."
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Cited by 7 (3 self)
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Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K5 or K3,3. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a lineartime decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE. 1
Constrained Simultaneous and Nearsimultaneous Embeddings
, 2007
"... A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ: V1 → V2 is a pair of planar straightline drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, wh ..."
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Cited by 7 (2 self)
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A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ: V1 → V2 is a pair of planar straightline drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the nearsimultaneous embedding problem, in which vertices are not forced to be placed exactly in the same, but just in “near” points in different drawings.
Crossing minimization meets simultaneous drawing
 In IEEE Pacific Visualisation Symposium
, 2008
"... We define the concept of crossing numbers for simultaneous graphs by extending the crossing number problem of traditional graphs. We discuss differences to the traditional crossing number problem, and give an NPcompleteness proof and lower and upper bounds for the new problem. Furthermore, we show ..."
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Cited by 7 (3 self)
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We define the concept of crossing numbers for simultaneous graphs by extending the crossing number problem of traditional graphs. We discuss differences to the traditional crossing number problem, and give an NPcompleteness proof and lower and upper bounds for the new problem. Furthermore, we show how existing heuristic and exact algorithms for the traditional problem can be adapted to the new task of simultaneous crossing minimization, and report on a brief experimental study of their implementations.
Colored simultaneous geometric embeddings
 In 13th Computing and Combinatorics Conference (COCOON
, 2007
"... Abstract. We introduce the concept of colored simultaneous geometric embeddings as a generalization of simultaneous graph embeddings with and without mapping. We show that there exists a universal pointset of size n for paths colored with two or three colors. We use these results to show that colore ..."
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Cited by 6 (3 self)
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Abstract. We introduce the concept of colored simultaneous geometric embeddings as a generalization of simultaneous graph embeddings with and without mapping. We show that there exists a universal pointset of size n for paths colored with two or three colors. We use these results to show that colored simultaneous geometric embeddings exist for: (1) a 2colored tree together with any number of 2colored paths and (2) a 2colored outerplanar graph together with any number of 2colored paths. We also show that there does not exist a universal pointset of size n for paths colored with five colors. We finally show that the following simultaneous embeddings are not possible: (1) three 6colored cycles, (2) four 6colored paths, and (3) three 9colored paths. 1
On a tree and a path with no geometric simultaneous embedding
 In Graph Drawing
, 2011
"... Two graphs G1 = (V,E1) and G2 = (V,E2) admit a geometric simultaneous embedding if there exist a set of points P and a bijection M: V → P that induce planar straightline embeddings both for G1 and for G2. The most prominent problem in this area is the question of whether a tree and a path can alway ..."
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Cited by 6 (1 self)
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Two graphs G1 = (V,E1) and G2 = (V,E2) admit a geometric simultaneous embedding if there exist a set of points P and a bijection M: V → P that induce planar straightline embeddings both for G1 and for G2. The most prominent problem in this area is the question of whether a tree and a path can always be simultaneously embedded. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edgedisjoint trees. Finally, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of height 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has height 4. Submitted:
Matched Drawings of Planar Graphs
, 2007
"... A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique ycoordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs all ..."
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Cited by 6 (3 self)
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A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique ycoordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3connected planar graphs or a 3connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of “carousel graphs”—are always matched drawable.
Simultaneous graph embedding with bends and circular arcs
 IN PROC. 14TH INTERN. SYMP. ON GRAPH DRAWING, VOLUME 4372 OF LNCS
, 2006
"... We consider the problem of simultaneous embedding of planar graphs. We demonstrate how to simultaneously embed a path and an nlevel planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph. We also show how to use starshaped levels to ..."
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Cited by 5 (3 self)
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We consider the problem of simultaneous embedding of planar graphs. We demonstrate how to simultaneously embed a path and an nlevel planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph. We also show how to use starshaped levels to find 2bends per path edge simultaneous embeddings of a path and an outerplanar graph. All embedding algorithms run in O(n) time.
An SPQRTree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges
, 2008
"... We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remai ..."
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Cited by 5 (0 self)
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We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQRtrees and has linear running time. We also show how we can employ SPQRtrees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQRtree approach might eventually lead to a polynomialtime algorithm for deciding the general SEFE problem for two planar graphs.