Intersection Graphs in Simultaneous Embedding with Fixed Edges

Jünger, Michael; Schulz, Michael

doi:10.7155/jgaa.00184

Cited by 29 publications

(27 citation statements)

References 8 publications

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“…Jünger and Schulz [6] show that two graphs admit a SEFE if and only if they admit planar embeddings that coincide on the intersection graph. In this sense, our obstructions give an understanding of which configurations should be avoided when looking for an embedding of the intersection graph.…”

Section: Introductionmentioning

confidence: 99%

A kuratowski-type theorem for planarity of partially embedded graphs

Jelínek

Kratochvíl

Rutter

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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A partially embedded graph (or Peg) is a triple (G, H, H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg (G, H, H) is planar if the graph G has a planar embedding that extends the embedding H.We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs.Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomial-time algorithm which, for a given Peg, either produces a planar embedding or identifies a minimal obstruction.

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Section: Introductionmentioning

confidence: 99%

A kuratowski-type theorem for planarity of partially embedded graphs

Jelínek

Kratochvíl

Rutter

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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“…Simultaneous Embedding Similarly, we can solve SEFE for two biconnected input graphs G 1 and G 2 whose intersection graph G is connected. Jünger and Schulz show that this is equivalent to the question whether embeddings of G 1 and G 2 exist that induce the same embedding for G [24,Theorem 4]. We start with the embedding representations D(G 1 ) and D(G 2 ).…”

Section: Constrained Embedding Problemsmentioning

confidence: 99%

“…Bläsius et al [4] give an extensive survey. Jünger and Schulz [24] show that two graphs admit a SEFE if and only if they have planar embeddings that coincide on the intersection graph.…”

Section: Constrained Embedding Problemsmentioning

confidence: 99%

Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems

Bläsius

Rutter

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

100

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In this paper, we define and study the new problem Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously, that is, for every child-parent relation the order chosen for the parent is an extension of the order chosen for the child. We show that Simultaneous PQ-Ordering is N Pcomplete in general and we identify a family of instances that can be solved efficiently, the 2-fixed instances. We show that this result serves as a framework for several other problems that can be formulated as instances of Simultaneous PQ-Ordering.In particular, we give linear-time algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a linear-time algorithm for Partially PQConstrained Planarity for biconnected graphs, which asks for a planar embedding in the presence of PQ-trees that restrict the possible orderings of edges around vertices, and a quadratic-time algorithm for Simultaneous Embedding with Fixed Edges for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two non-trivial blocks. This includes for example the case where both graphs have maximum degree 5.

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“…Simultaneous embeddings are desirable for representing pairs of related planar graphs and have applications in graph visualization and graph drawing, The problem of deciding whether two graphs have a SEFE is open in the general case but there is an efficient algorithm for the case when the shared graph is 2-connected [18,1]. Several results are known for restricted pairs of graphs [20,10,14,9,13,12,1,18].…”

Section: Introductionmentioning

confidence: 99%

The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs

Jampani

Lubiw

2009

Lecture Notes in Computer Science

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We introduce a notion of simultaneity for any class of graphs with an intersection representation (interval graphs, chordal graphs, etc.) and for comparability graphs, which are represented by transitive orientations. Let G 1 and G 2 be graphs from such a class C, sharing some vertices I and the corresponding induced edges. Then G 1 and G 2 are said to be simultaneous C graphs if there exist representations R 1 and R 2 of G 1 and G 2 that are the same on the shared subgraph.Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip, etc.For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G 1 and G 2 , sharing vertices I and the corresponding induced edges, do there exist edges E ⊆ V (G 1 ) − I × V (G 2 ) − I such that the graph G 1 ∪ G 2 ∪ E belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to some well-studied classes in the literature, namely, sandwich graphs and probe graphs.We give efficient algorithms for solving the simultaneous representation problem for chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs. Finally, we show that the general version of the problem is NP-hard for chordal graphs.

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Intersection Graphs in Simultaneous Embedding with Fixed Edges

Cited by 29 publications

References 8 publications

A kuratowski-type theorem for planarity of partially embedded graphs

A kuratowski-type theorem for planarity of partially embedded graphs

Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems

The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs

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