In a graph G, a vertex is said to dominate itself and all vertices adjacent to it. For a positive integer k, the k-tuple domination number γ×k(G) of G is the minimum size of a subset D of V (G) such that every vertex in G is dominated by at least k vertices in D. To generalize/improve known upper bounds for the k-tuple domination number, this paper establishes that for any positive integer k an...

Let k ≥ 1 be an integer and G a graph of minimum degree δ ( ) − 1. A set D ⊆ V is said to -tuple dominating if | N [ v ] ∩ for every vertex ∈ ), where represents the closed neighbourhood . The cardinality among all sets domination number In this paper, we continue with study classical parameter in graphs. particular, provide some relationships that exist between other parameters, like multiple ...

‎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, ...

A set D ⊆ V is called a k-tuple dominating set of a graph G = (V,E) if |NG[v] ∩D| ≥ k for all v ∈ V , where NG[v] denotes the closed neighborhood of v. A set D ⊆ V is called a liar’s dominating set of a graph G = (V,E) if (i) |NG[v] ∩D| ≥ 2 for all v ∈ V , and (ii) for every pair of distinct vertices u, v ∈ V , |(NG[u] ∪NG[v]) ∩D| ≥ 3. Given a graph G, the decision versions of k-Tuple Dominatio...

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...