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 almost one path in ψ and every edge of G is an exactly one path in ψ. If further no member of ψ is a cycle, then ψ is called an acyclic graphoidal cover of G. The minimum cardinality of an acyclic graphoidal cover is called the acyclic graphoidal covering number of G is denoted by η a. In this paper we characterize the class of graphs for which η a = q - p where p and q denote respectively the order and size of G.