On the Paired-Domination Subdivision Number of a Graph

Guoliang Hao,Rana Khoeilar,Hossein Karami,Mustapha Chellali,Seyed Mahmoud Sheikholeslami

doi:10.3390/math9040439

Abstract

In order to increase the paired-domination number of a graph G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number sdγpr(G) of G. It is well known that sdγpr(G+e) can be smaller or larger than sdγpr(G) for some edge e∉E(G). In this note, we show that, if G is an isolated-free graph different from mK2, then, for every edge e∉E(G), sdγpr(G+e)≤sdγpr(G)+2Δ(G).

Highlights

All graphs considered in this paper are finite, simple, and undirected
The open neighborhood NG (v) of a vertex v in G is the set of all vertices that are adjacent to v, the closed neighborhood
We provide an upper bound for sdγ pr ( G + e) for any e ∈

Summary

Introduction

All graphs considered in this paper are finite, simple, and undirected. Let V ( G ) and. The minimum cardinality of a PD-set of G is called the paired-domination number of G and is denoted by γ pr ( G ). Fink et al [9] introduced the bondage number of a graph, which is the minimum number of edges in which removal increases the domination number. In order to increase the paired-domination number of G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number and is denoted by sdγ pr ( G ). We note that the subdivision of the unique edge of a path of order 2 does not increase the paired-domination number. Let St (t ≥ 4) denote the subdivided star obtained from a star K1,t−1 of order t by subdividing all edges of K1,t−1.

Preliminary Results

Proof of Theorem 1

Conclusions and Open Problems