The length of dominating cycle of claw-free graph

Xiaodong Chen,Lu Zhi,Meijin Xu

doi:10.1177/1687814017737726

Abstract

The length of dominating cycles is usually discussed in control problems. A dominating cycle of a graph G is a cycle C of G such that V(G)−V(C) is an independent set. In this article, we prove that for any claw-free graph G with δ(G)≥2, the length of longest dominating cycle is at least min{n,2|NC2(G)|−1}, where NC2(G) denotes the vertex set N(u)∪N(v) containing the minimum number of vertices for all vertices u,v with d(u,v)=2 in G.

Highlights

The length of dominating cycles in graphs is usually discussed in control problems.[1,2,3,4,5,6,7,8,9]
Theorem 1.4.13 If G contains a dominating cycle with order n and d(G) ! 2, G contains a dominating cycle of length at least minfn, 2jNC2(G)j À 3g
If G is a claw-free graph of order n containing a dominating cycle with d(G) ! 2, the length of a longest dominating cycle is at least minfn, 2jNC2(G)j À 1g

Summary

Introduction

The length of dominating cycles in graphs is usually discussed in control problems.[1,2,3,4,5,6,7,8,9] In this article, we only consider finite and simple graphs, and for the notation and terminology not defined here, please refer to Bondy and Murty[10] The length of a longest cycle in a graph G is called the circumference of G. N + 2, the circumference of G is at least minfn, 2jNC2(G)jg, where s3(G) denotes the minimum sum degree of all three independent vertices in G. Theorem 1.4.13 If G contains a dominating cycle with order n and d(G) ! N + 2, the circumference of G is at least minfn, 2jNC(G)jg, where s3(G) denotes the Corresponding author: Xiaodong Chen, College of Science, Liaoning University of Technology, Liaoning, Jinzhou 121001, P.R. China.

Results

Conclusion