Strong total domination and weak total domination in Mycielski’s graphs

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DOI:

https://doi.org/10.26637/MJM0804/0060

Abstract

Let G=(V,E) be a graph. A set SV is called a weak total dominating set (WTD-set) if each vertex vVS is adjacent to a vertex uS with deg(v)>deg(u) and every vertex in S adjacent to a vertex in S. The weak total domination number, denoted by γut(G), is minimum cardinality of a weak total dominating set. Anologuosly, a dominating set SV is called a strong total dominating set (STD-set) if each vertex vVS is dominated by some vertices uS with deg(v)<deg(u) and each vertex in S adjacent to a vertex in S. The strong total domination number, denoted by γs(G), is minimum cardinality of a strong dominating set. Weak total and strong total domination parameters were introduced by Chellali et al. and Akbari and Jafari Rad, respectively.
In this paper, we consider weak total and strong total domination of Mycielski's Graph, denoted by μ(G). We also provide some upper and lower bound about weak total domination of Mycielski's graph related with minimum and maximum degree number of a graph. In addition, the inequality about relationship between strong total domination of Mycieski's graph μ(G) and underlying graph G,γst(G)+1γst(μ(G))γst(G)+2, is obtained. Among other results, we characterize graphs G achieving the lower bound γst(G)+1=γst(μ(G)).

Keywords:

Graph Theory, Strong Total Domination,, Weak Total Domination, Mycielski’s Graph

Mathematics Subject Classification:

Mathematics
  • Pages: 1681-1686
  • Date Published: 01-10-2020
  • Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

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Published

01-10-2020

How to Cite

Hande Tunc¸el G¨olpek, and Aysun Aytac¸. “Strong Total Domination and Weak Total Domination in Mycielski’s Graphs”. Malaya Journal of Matematik, vol. 8, no. 04, Oct. 2020, pp. 1681-6, doi:10.26637/MJM0804/0060.