The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs - Jampani, Krishnam Raju; Lubiw, Anna (Journal of Graph Algorithms and Applications, 2012)
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 G1 and G2 be graphs from such a classC,sharing some vertices I and the corresponding induced edges. ThenG1 and G2 are said to be simultaneous Cgraphs if there exist representations R1 and R2 ofG1 and G2 that are the sameon the shared subgraph. <!----> Simultaneous representation problems arise in any situation where two related graphsshould be represented consistently. A main instance is for temporal relationships, where an old graph and a new graphshare some common parts. Pairs of related graphs arise in many other situations. For example, two social networks thatshare 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 graphclass is equivalent to a graph augmentation problem: given graphs G1 and G2, sharing vertices I and the correspondinginduced edges, do there exist edges E′ ⊆ V(G1)−I ×V(G2)−I such that the graph G1 ∪G2 ∪E′ belongsto the graph class. This equivalence implies that thesimultaneous representation problem is closely related to some well-studied classes in the literature, namely, sandwich graphsand 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 amore general version of the problem when there are multiple graphs, any two of which share the same common graph. Thisversion of the problem also generalizes probe graphs. Finally, we show that the general version of the problem is NP-hard for chordalgraphs. <!---->