EQUALITY OF GRAPHOIDAL AND ACYCLIC GRAPHOIDAL COVERING NUMBER OF A GRAPH

A 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 acyclic graphoidal cover of G. The minimum cardinality of a graphoidal cover is called the graphoidal covering number of G and is denoted by 77 and the minimum cardinality of an acyclic graphoidal cover is called an 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).