Induced Graphoidal Decompositions in Product Graphs

Abstract

Let be a nontrivial, simple, finite, connected, and undirected graph. A graphoidal decomposition (GD) of is a collection of nontrivial paths and cycles in that are internally disjoint such that every edge of lies in exactly one member of . By restricting the members of a GD to be induced, the concept of induced graphoidal decomposition (IGD) of a graph has been defined. The minimum cardinality of an IGD of a graph is called the induced graphoidal decomposition number and is denoted by (). An IGD of without any cycles is called an induced acyclic graphoidal decomposition (IAGD) of , and the minimum cardinality of an IAGD of is called the induced acyclic graphoidal decomposition number of , denoted by (). In this paper we determine the value of () and () when is a product graph, the factors being paths/cycles.