Acyclic graphoidal covers and path partitions in a graph

Abstract

An acyclic graphoidal cover of a graph G is a collection ψ of paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering number of G and is denoted by ηa. A path partition of a graph G is a collection script P sign of paths in G such that every edge of G is in exactly one path in script P sign. The minimum cardinality of a path partition of G is called the path partition number of G and is denoted by π. In this paper we determine ηa and π for several classes of graphs and obtain a characterization of all graphs with Δ≤4 and ηa = Δ - 1. We also obtain a characterization of all graphs for which ηa = π. © 1998 Elsevier Science B.V. All rights reserved.