# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## On the Total Restrained Domination Number of Direct Products of Graphs

 Wai Chee Shiu Department of Mathematics, Hong Kong Baptist University 224 Waterloo Road, Kowloon Tong, Hong Kong, China Hong-Yu Chen School of Mathematics and System Sciences, Shandong University Jinan, Shandong Province, 250100, China Xue-Gang Chen Department of Mathematics, North China Electric Power University Beijing, 102206, China Pak Kiu Sun Department of Mathematics, Hong Kong Baptist University 224 Waterloo Road, Kowloon Tong, Hong Kong, China

## 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 γrt(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.

Keywords: total domination number, total restrained domination number, direct product of graphs

2010 Mathematics Subject Classification: 05C69.

## References

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