On the Paired-Domination Subdivision Number of Trees

Shouliu Wei,Seyed Mahmoud Sheikholeslami,Hossein Karami,Rana Khoeilar,Guoliang Hao

doi:10.3390/math9101135

Abstract

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T ≠ P5 of order n ≥ 3 and each edge e ∉ E(T), sdγpr(T) + sdγpr(T + e) ≤ n + 2.

Highlights

Let G = (V, E) be a simple connected graph with vertex set V = V ( G ) and edge setE( G ) = E and let n = |V |
Velammal [8] was the first to study the domination subdivision number of a graph G defined to be the minimum number of edges that must be subdivided to increase the domination number
Let G be a connected graph of order n ≥ 3 and let G 0 be obtained from G by subdividing the edge e = uv ∈ E( G )

Summary

Introduction

Let G = (V, E) be a simple connected graph with vertex set V = V ( G ) and edge set. The paireddomination subdivision number sdγ pr ( G ) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the paired-domination number of G. Our aim in this paper is to further study paired-domination subdivision number and show that for each tree T 6= P5 of order n ≥ 3 and each edge e 6∈ E( T ), sdγ pr ( T ) +. Let T2,m be the tree obtained from T1,m by subdividing the edge xy with a subdivision vertex u and adding a new vertex v and a new edge uv (Figure 3). With equality if and only if T ∈ {K1,3 , T1,m , T2,m | m ≥ 1}

Main Result

Conclusions