Helly Property and Sandwich Graphs

The purpose of this paper consists in analysing the recognizing problems of the clique-Helly graphs and the hereditary clique-Helly graphs in sandwich version. Let F be a family of subsets of a set S. We say that F satisfies the Helly property when every subfamily consisting of pairwise intersecting subsets has a non-empty intersection. A graph is clique-Helly when its family of maximal cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. A graph G(V, E) is a sandwich graph for the pair G1(V, E1), G2(V, E2), such that E1 ⊆ E ⊆ E2. A graph sandwich problem consists in, given two graphs G1 e G2, finding a sandwich graph G with a property Π. Graph sandwich problems were defined in the context of Computational Biology and have many applications and it is a natural generalization of recognizing problems [1]. Only problems with polynomial time recognizing are interesting in its sandwich version.