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.

In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. Say that a family of subsets has the Helly property when every subfamily of it, formed by pairwise intersecting subsets, contains a common element. There are many generalizations of this property which are relevant to some parts of mathematics and several applications in computer science. In this work, we survey computational aspects of the Helly property. The main focus is algorithmic. That is, we describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NP-hardness results.

The Helly defect of a graph G is the smallest integer i such that the iterated clique graph Ki(G) is clique-Kelly. We prove that it is NP-hard to decide whether the Helly defect of G is at most 1.