Table of Contents
ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 806193, 22 pages
http://dx.doi.org/10.5402/2011/806193
Research Article

Equivalence between Hypergraph Convexities

Computer Science Department, Sapienza University of Rome, 00185 Rome, Italy

Received 25 August 2011; Accepted 29 September 2011

Academic Editors: G. Isaak and J. Tarhio

Copyright © 2011 Francesco M. Malvestuto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 be a connected graph on 𝑉. A subset 𝑋 of 𝑉 is all-paths convex (or ap-convex) if 𝑋 contains each vertex on every path joining two vertices in 𝑋 and is monophonically convex (or π‘š-convex) if 𝑋 contains each vertex on every chordless path joining two vertices in 𝑋. First of all, we prove that ap-convexity and π‘š-convexity coincide in 𝐺 if and only if 𝐺 is a tree. Next, in order to generalize this result to a connected hypergraph 𝐻, in addition to the hypergraph versions of ap-convexity and π‘š-convexity, we consider canonical convexity (or 𝑐-convexity) and simple-path convexity (or sp-convexity) for which it is well known that π‘š-convexity is finer than both 𝑐-convexity and sp-convexity and sp-convexity is finer than ap-convexity. After proving sp-convexity is coarser than 𝑐-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of 𝛾-acyclic hypergraphs.