On the total restrained domination number of direct products of graphs

Wai Chee Shiu; Hong-Yu Chen; Xue-Gang Chen; Pak Kiu Sun

doi:10.7151/dmgt.1632

On the total restrained domination number of direct products of graphs

Wai Chee Shiu^*, Hong-Yu Chen, Xue-Gang Chen, Pak Kiu Sun

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

1 Citation (Scopus)

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Abstract

Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V \ S is adjacent to a vertex in S as well as to another vertex in V \ S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ ^t_r(G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

Original language	English
Pages (from-to)	629-641
Number of pages	13
Journal	Discussiones Mathematicae - Graph Theory
Volume	32
Issue number	4
DOIs	https://doi.org/10.7151/dmgt.1632
Publication status	Published - Dec 2012

Scopus Subject Areas

Discrete Mathematics and Combinatorics
Applied Mathematics

User-Defined Keywords

total domination number
total restrained domination number
direct product of graphs

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AU - Chen, Hong-Yu

AU - Chen, Xue-Gang

AU - Sun, Pak Kiu

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N2 - Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V \ S is adjacent to a vertex in S as well as to another vertex in V \ S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ tr(G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

AB - Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V \ S is adjacent to a vertex in S as well as to another vertex in V \ S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ tr(G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

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On the total restrained domination number of direct products of graphs

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