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

Abstract

Let \(G=(V, E)\) be a graph. A set \(S \subseteq V\) is called a weak total dominating set (WTD-set) if each vertex \(v \in V-S\) is adjacent to a vertex \(u \in S\) with \(\operatorname{deg}(v)>\operatorname{deg}(u)\) and every vertex in \(S\) adjacent to a vertex in \(S\). The weak total domination number, denoted by \(\gamma_{\mathrm{ut}}(G)\), is minimum cardinality of a weak total dominating set. Anologuosly, a dominating set \(S \subseteq V\) is called a strong total dominating set (STD-set) if each vertex \(v \in V-S\) is dominated by some vertices \(u \in S\) with \(\operatorname{deg}(v)<\operatorname{deg}(u)\) and each vertex in \(S\) adjacent to a vertex in \(S\). The strong total domination number, denoted by \(\gamma_s(G)\), 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 \(\mu(G)\). 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 \(\mu(G)\) and underlying graph \(G, \gamma_{s t}(G)+1 \leq \gamma_{s t}(\mu(G)) \leq \gamma_{s t}(G)+2\), is obtained. Among other results, we characterize graphs \(G\) achieving the lower bound \(\gamma_{s t}(G)+1=\gamma_{s t}(\mu(G))\).