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