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 ηaand 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.
Arumugam, S., & Suresh Suseela, J. (1998). Acyclic graphoidal covers and path partitions in a graph. Discrete Mathematics, 190(1–3), 67–77. https://doi.org/10.1016/S0012-365X(98)00032-6