The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs

Abstract

We introduce the simultaneous representation problem, defined for any graph class \(\cal C\) characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G 1 and G 2, sharing some vertices X (and the corresponding induced edges), are said to have a simultaneous representation with respect to a graph class \(\cal C\), if there exist representations R 1 and R 2 of G 1 and G 2 that are “consistent” on X. Equivalently (for the classes \(\cal C\) that we consider) there exist edges E′ between G 1 − X and G 2 − X such that G 1 ∪ G 2 ∪ E′ belongs to class \(\cal C\).Simultaneous representation problems are related to graph sandwich problems, probe graph recognition problems and simultaneous planar embeddings and have applications in any situation where it is desirable to consistently represent two related graphs.In this paper we give efficient algorithms for the simultaneous representation problem on chordal, comparability and permutation graphs. These results complement the recent poly-time algorithms for recognizing probe graphs for the above classes and imply that the graph sandwich problem for these classes is solvable for an interesting special case: when the set of optional edges induce a complete bipartite graph. Moreover for comparability and permutation graphs, our results can be extended to solve a generalized version of the simultaneous representation problem when there are k graphs any two of which share a common vertex set X. This generalized version is equivalent to the graph sandwich problem when the set of optional edges induce a k-partite graph.KeywordsSimultaneous graphsSandwich graphsChordal graphsComparability graphsPermutation graphs