Equality of graphoidal and acyclic graphoidal covering number of a graph

Abstract

A {\it graphoidal cover} of a graph $ G $ is a collection $ \psi $ of (not necessarily open) paths in $ G $ such that every vertex of $ G $ is an internal vertex of at most one path in $ \psi $ ad every edge of $ G $ is in exactly one path in $ \psi $. If no member of $ \psi $ is a cycle, then $ \psi $ is called an {\it acyclic graphoidal cover} of $ G $. The minimum cardinality of a graphoidal cover is called the {\it graphoidal covering number} of $ G $ and is denoted by $ \eta $ and the minimum cardinality of an acyclic graphoidal cover is called an {\it acyclic graphoidal covering number} of $ G $ and is denoted by $ \eta_a $. In this paper we characterize the class of graphs for which $ \eta=\eta_a $.