Simple Graphoidal Covers in a Graph

A simple graphoidal cover of a graph \( G \) is a collection \( \psi \) of (not necessarily open) paths in \( G \) such that every path in \( \psi \) has at least two vertices, every vertex of \( G \) is an internal vertex of at most one path in \( \psi \), every edge of \( G \) is in exactly one path in \( \psi \), and any two paths in \( \psi \) have at most one vertex in common. The minimum cardinality of a simple graphoidal cover of \( G \) is called the simple graphoidal covering number of \( G \) and is denoted by \( \eta_s(G) \). In this paper, we determine the value of \( \eta_s \) for several families of graphs. We also obtain several bounds for \( \eta_s \) and characterize graphs attaining the bounds.