Abstract 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 atmost one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G and is denoted by η. In this paper we characterize the class of graphs with edge connectivity one and η = q − p where p and q denote the order and size of G respectively.