On a conjecture involving a bound for the total restrained domination number of a graph

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2 Citations (Scopus)

Abstract

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S, and every vertex in V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted γ tr (G), is the smallest cardinality of a total restrained dominating set of G. In Koh et al. (2013), it is proved that if G is a graph of order n≥4 and δ(G)≥2, then γ tr (G)≤n−[Formula presented]3. It is further conjectured that this bound can be improved to γ tr (G)≤n−θ(n). In this paper we show that if G is a graph with no C 3 components and δ(G)≥2, then γ tr (G)≤n−[Formula presented].

Original languageEnglish
Pages (from-to)177-187
Number of pages11
JournalDiscrete Applied Mathematics
Volume258
DOIs
Publication statusPublished - 15 Apr 2019

Keywords

  • Domination
  • Graph
  • Total restrained domination
  • Upper bound

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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