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

Abstract

A family C of sets has the Helly property if any subfamily C ′ whose elements are pairwise intersecting has non-empty intersection. Suppose that C is a non-empty family of subsets of a finite set V : the Helly number h ( C ) of C is the least positive integer n such that every n -wise intersecting subfamily of C has non-empty intersection. In this paper the Helly property of families of convex sets relative to two new graph convexities are studied. Let G be a (finite) connected graph and U a set of vertices of G . A connected subgraph with the fewest 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 g 3 - 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 set of all vertices that belong to some minimal U -tree is called the monophonic interval for U and a set S of vertices is m 3 - convex if it contains the monophonic interval of every 3 -subset U of S . Those graphs are characterized for which the families of g 3 -convex ( m 3 -convex) sets of size at least 3 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 . The near-clique number of a graph is a lower bound on the Helly number for both g 3 -convex families and m 3 -convex families. For m 3 -convex families equality holds. For g 3 -convex families equality holds for chordal graphs and for distance-hereditary graphs, but the bound can be arbitrarily bad in general, even when the near-clique number is 3 .