A General Method for Forbidden Induced Subgraph Sandwich Problem NP-completeness

Abstract

We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic and Shamir (1993), with respect to classes of graphs defined by excluding induced subgraphs. The Π graph sandwich problem asks, for a pair of graphs G1 = (V, E1) and G2 = (V, E2) with E1 ⊆ E2, whether there exists a graph G = (V, E) with E1 ⊆ E ⊆ E2 that satisfies property Π. We consider the property of being H-free, where H is a fixed graph. Using a new variant of the SAT problem, we present a general framework to establish the NP-completeness of the sandwich problem for several H-free graph classes which generalizes the previous strategy for the class of Hereditary clique-Helly graphs. We also provide infinite families of 3-connected special forbidden induced subgraphs for which each forbidden induced subgraph sandwich problem is NP-complete.