Bounds on Co-Independent Liar’s Domination in Graphs

K Suriya Prabha,S Amutha,Ismail Naci Cangul,N Anbazhagan,Ghulam Shabbir

doi:10.1155/2021/5544559

Abstract

A set S ⊆ V of a graph G = V , E is called a co-independent liar’s dominating set of G if (i) for all v ∈ V , N G v ∩ S ≥ 2 , (ii) for every pair u , v ∈ V of distinct vertices, N G u ∪ N G v ∩ S ≥ 3 , and (iii) the induced subgraph of G on V − S has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar’s domination number of G , and it is denoted by γ coi L R G . In this paper, we introduce the concept of co-independent liar’s domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter.

Highlights

For notations and nomenclature, we refer [1]
Let G (V, E) be a graph with vertex set V of order p |V| and edge set E of size q |E|. e diameter of G is the greatest distance between any two vertices of G. e middle graph M(G) is the derived graph obtained from G by inserting a new vertex into every edge of G and joining these new vertices by edges which lie on the adjacent edges of G [2]
A topological index is a real number related to a graph, which must be a structural invariant. e topological indices are a vital tool for quantitative structure activity relationship and quantitative structure property relationship

Summary

Introduction

Let G (V, E) be a graph with vertex set V of order p |V| and edge set E of size q |E|. E diameter of G is the greatest distance between any two vertices of G. e middle graph M(G) is the derived graph obtained from G by inserting a new vertex into every edge of G and joining these new vertices by edges which lie on the adjacent edges of G [2]. Haynes et al introduced the concept of domination in graphs [3]. A topological index is a real number related to a graph, which must be a structural invariant. E topological indices are a vital tool for quantitative structure activity relationship and quantitative structure property relationship. For more work on topological indices of a graph, refer recent papers [4, 5]

Co-Independent Liar’s Domination in Graphs

Conclusion