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33
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 28 (11 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Simultaneous Geometric Graph Embeddings
"... 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 pr ..."
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Cited by 23 (7 self)
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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.
Simultaneous graph embeddings with fixed edges
 IN 32ND WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE (WG
, 2006
"... 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 ..."
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Cited by 18 (9 self)
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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.
On a tree and a path with no geometric simultaneous embedding
, 2012
"... 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 10 (2 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.
DeltaConfluent Drawings
, 2005
"... We generalize the treeconfluent graphs to a broader class of graphs called ∆confluent graphs. This class of graphs and distancehereditary graphs, a wellknown class of graphs, coincide. Some results about the visualization of ∆confluent graphs are also given. ..."
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Cited by 10 (2 self)
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We generalize the treeconfluent graphs to a broader class of graphs called ∆confluent graphs. This class of graphs and distancehereditary graphs, a wellknown class of graphs, coincide. Some results about the visualization of ∆confluent graphs are also given.
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 9 (2 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.
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 9 (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 9 (3 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.