BONDAGE AND STRONG-WEAK BONDAGE NUMBERS OF TRANSFORMATION GRAPHS $G^{xyz}$

A Ayta{\C{C}},T Turaci

doi:10.12732/ijpam.v106i2.30

Abstract

Let G(V (G),E(G)) be a simple undirected graph. A dominating set of G is a subset D ⊆ V (G) such that every vertex in V (G) −D is adjacent to at least one vertex in D. The minimum cardinality taken over all dominating sets of G is called the domination number of G and also is denoted by (G). There are a lot of vulnerability parameters depending upon dominating set. These parameters are strong and weak domination numbers, reinforcement number, bondage number, strong and weak bondage numbers, etc. The bondage parameters are important in these parameters. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than (G). In this paper, the bondage parameters have been examined of transformation graphs, then exact values and upper bounds have been obtained.

Highlights

In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links
If we think of a graph as modeling a network, there are many graph theoretical parameters such as domination number, strong and weak domination numbers, bondage number, strong and weak bondage numbers
The complement G of a graph G has V (G) as its vertex sets, but two vertices are adjacent in G if only if they are not adjacent in G

Summary

Introduction

The vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. The complement G of a graph G has V (G) as its vertex sets, but two vertices are adjacent in G if only if they are not adjacent in G (see [5]-[13]). The strong bondage number of G, as the minimum cardinality among all sets of edges E′ ⊆ E(G) such that γs(G − E′) > γs(G) and it is denoted by bs(G).

Results

Conclusion