Abstract

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

Highlights

  • In this paper, G is a simple nontrivial connected graph with vertex set V = V ( G ) and edge setE = E( G )

  • The order |V | of G is denoted by n = n( G )

  • If v is a support vertex, Lv denotes the set of leaves adjacent to v

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Summary

Introduction

To ve-dominate the edge uv, we may assume that D ∩ ( N [u] − {v}) 6= ∅ and D is clearly a dve-dominating set of G implying that γdve ( G 0 ) ≥ γdve ( G ). If u ∈ WG2 , clearly some γdve ( G )-set contains a vertex NG [u], and so it can be extended to a dve-dominating set of G 0 by adding x2 , x3 and γdve ( G 0 ) ≤ γdve ( G ) + 2.

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