Simple Path Covers in Graphs

A simple path cover of a graph G is a collection ψ of paths in G such that every edge of G is in exactly one path in ψ and any two paths in ψ have at most one vertex in common. More generally, for any integer k ≥ 1, a Smarandache path k-cover of a graph G is a collection ψ of paths in G such that each edge of G is in at least one path of ψ and two paths of ψ have at most k vertices in common. Thus if k = 1 and every edge of G is in exactly one path in ψ, then a Smarandache path k-cover of G is a simple path cover of G. The minimum cardinality of a simple path cover of G is called the simple path covering number of G and is denoted by πs(G). In this paper we initiate a study of this parameter.