Abstract
Abstract A sandwich problem for property π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property π Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is NP-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all NP-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.
Highlights
INTRODUCTIONA polynomial-time recognition algorithm for the class of hereditary clique-Helly graphs was presented in [13]
This paper is organized as follows: in Section 2 we prove that HEREDITARY CLIQUE-HELLY SANDWICH PROBLEM, CLIQUE-HELLY SANDWICH PROBLEM, and CLIQUE GRAPH SANDWICH PROBLEM are N P -complete; in Section 3 we prove that CLIQUEHELLY NONHEREDITARY SANDWICH PROBLEM is N P complete; in Section 4 we have our concluding remarks about complementary graph classes and sandwich problems
We turn our attention to the CLIQUE GRAPH SANDWICH PROBLEM and we present a simple proof of its N P -completeness
Summary
A polynomial-time recognition algorithm for the class of hereditary clique-Helly graphs was presented in [13] Every studied sandwich problem corresponding to a graph class defined by a finite family of minimal forbidden subgraphs has been classified as Polynomial. The class hereditary clique-Helly is the first example of a graph class defined by a finite family of minimal forbidden subgraphs for which the corresponding sandwich problem is N P -complete. It is clear that G1 and G2 contain, respectively, a clause Hajós graph and a variable Hajós graph, G1 and G2 are not hereditary clique-Helly. In the hereditary clique-Helly sandwich graph G, for each forced clause Hajós graph, we must have at least one added optional edge. This central triangle is not in a K4 because there is no vertex adjacent in G2 to these three vertices
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