Helly theorems for 3-Steiner and 3-monophonic convexity in graphs

A family C of sets has the Helly property if any subfamily C′, whose elements are pairwise intersecting, has non-empty intersection. Suppose C is a non-empty family of subsets of a finite set V . The Helly number h(C) of C is the smallest positive integer n such that every subfamily C′ of C with |C′| ≥ n and which intersects n-wise has non-empty intersection. In this paper we consider the families of convex sets relative to two graph convexities. Suppose G is a (finite) connected graph and U a set of vertices of G. Then a connected subgraph with the fewest number of edges containing U is called a Steiner tree for U , and the collection of all vertices of G that belong to some Steiner tree for U is called the Steiner interval for U . A set S of vertices of G is g3-convex if it contains the Steiner interval for every 3-subset U of S. A subtree T of G that contains U is a minimal U -tree if every vertex of T that is not in U is a cut-vertex of the subgraph induced by V (T ). The collection of all vertices that belong to some minimal U -tree is called the monophonic interval for U and a set S of vertices is m3-convex if it contains the monophonic interval of every 3-subset U of S. We characterize those (finite) graphs for which the families of convex sets, of cardinality at least 3, with respect to the g3-convexity and m3convexity have the Helly property. A graph obtained from a complete graph by deleting a matching is called a near-clique. The maximum order of a near-clique in a graph G is called the near-clique number of G. It is observed that the near-clique number of a graph is a lower bound on the Helly number for both the family of g3and m3-convex sets. It is shown that the near-clique number of chordal and distance-hereditary graphs equals the Helly number of the g3-convex sets for these graphs and it is shown that there are graphs with near-clique number 3 for which the Helly number of the g3-convex sets is arbitrarily large. For the m3-convex sets it is shown that the near-clique number always equals the Helly number.