‎let $g=(v(g),e(g))$ be a graph‎, ‎$gamma_t(g)$. let $ooir(g)$ be the total domination and oo-irredundance number of $g$‎, ‎respectively‎. ‎a total dominating set $s$ of $g$ is called a $textit{total perfect code}$ if every vertex in $v(g)$ is adjacent to exactly one vertex of $s$‎. ‎in this paper‎, ‎we show that if $g$ has a total perfect code‎, ‎then $gamma_t(g)=ooir(g)$‎. ‎as a consequence, ...

We consider four different types of multiple domination and provide new improved upper bounds for the kand k-tuple domination numbers. They generalise two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets...

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

In combinatorics, a latin square is $$n\times n$$ matrix filled with n different symbols, each occurring exactly once in row and column. Associated to square, we can define simple graph called graph. this article, compute lower upper bounds for the domination number k-tuple total numbers of such graphs. Moreover, describe formula 2-tuple number.

The following fundamental result for the domination number γ(G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ(G) ≤ ln(δ + 1) + 1 δ + 1 n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [On double domination in graphs. Discuss. Math. Graph Theory 25 (2005) 29–34...