Induced Graphoidal Decompositions in Product Graphs

Mayamma Joseph,I Sahul Hamid

doi:10.1155/2013/892839

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.

Highlights

By a graph G = (V, E) we mean a nontrivial, finite, connected, and undirected graph having no loops and multiple edges
Harary [2] introduced the concept of path decomposition of graphs in 1970 which was further studied by Harary and Schwenk [3], Peroche [4], and Stanton et al [5]
As a special case of path decomposition, Acharya and Sampathkumar [6] introduced the notion of graphoidal decomposition which is a decomposition of a graph into internally disjoint paths/cycles

Summary

Introduction

By imposing the condition that the members of a graphoidal decomposition are induced paths/cycles, Arumugam [7] introduced the concept of induced graphoidal decomposition as well as that of induced acyclic graphoidal decomposition. Studies on these decompositions were initiated by Ratan Singh and Das [8, 9] and were further extended by Sahul Hamid and Joseph [10, 11] by obtaining certain bounds of the related parameters ηi and ηia and solving some characterization problems. In this paper we determine the value of ηi and ηia for a class of product graphs, namely, products of paths and cycles

Induced Graphoidal Decomposition

Induced Graphoidal Decomposition Number of Products of Paths and Cycles

Scope for Further Research