The Simultaneous Membership Problem for Chordal, Comparability and   Permutation graphs
  In this paper we introduce the 'simultaneous membership problem', defined for any graph class 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 be 'simultaneous members' of graph class 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 C that we consider) there exist edges E' between G_1-X and G_2-X such that G_1 \cup G_2 \cup E' belongs to class C.   Simultaneous membership problems have application in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated version of the other. Simultaneous membership problems are related to simultaneous planar embeddings, graph sandwich problems and probe graph recognition problems.   In this paper we give efficient algorithms for the simultaneous membership problem on chordal, comparability and permutation graphs. These results imply that graph sandwich problems for the above classes are tractable for an interesting special case: when the set of optional edges form a complete bipartite graph. Our results complement the recent polynomial time recognition algorithms for probe chordal, comparability, and permutation graphs, where the set of optional edges form a clique. 
Krishnam Raju Jampani and Anna Lubiw