Decomposable Convexities in Graphs and Hypergraphs

Abstract

Given a connected hypergraph ℋwith vertex set V, a convexity space on ℋ is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in ℋ. e members of are called convex sets. e convex hull of a subset X of V is the smallest convex set containing X. By a cluster of ℋ we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on ℋ is decomposable if it satis�es the following three axioms: (i) the maximal clusters of ℋ form an acyclic hypergraph, (ii) every maximal cluster of ℋ is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on ℋ is fully speci�ed by the maximal clusters of ℋ in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of ℋ and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of ℋ are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as "monophonic" and "canonical" convexities in hypergraphs and "all-paths" convexity in graphs).

A Matrix Approach to Hypergraph Stable Set and Coloring Problems with Its Application to Storing Problem

“…The coloring problems of a special kind of graphs have been widely discussed in [7][8][9]. In recent years, there have been some references considering hypergraph theory, such as [10,11]. It has been successfully applied to many different areas such as Markov decision process [12], complete simple games [13], linear programming [14], and cooperation structures in games [15].…”