On Graphs Whose Acyclic Graphoidal Covering Number Is One Less than Its Cyclomatic Number

Abstract

A graphoidal cover of a graph \(G\) is a collection \(\psi\) of (not necessarily open) paths in \(G\) such that every vertex of \(G\) is an internal vertex of at most one path in \(\psi\) and every edge of \(G\) is an exactly one path in \(\psi\). If further no member of \(\psi\) is a cycle, then \(\psi\) is called an acyclic graphoidal cover of \(G\). The minimum cardinality of an acyclic graphoidal cover is called the acyclic graphoidal covering number of \(G\) and is denoted by \(\eta_a\). In this paper, we characterize the class of graphs for which \(\eta_a = q – p\), where \(p\) and \(q\) denote respectively the order and size of \(G\).