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 ...

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.

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...

A dominating set S of a graph G is called efficient if |N [v]∩S| = 1 for every vertex v ∈ V (G). That is, a dominating set S is efficient if and only if every vertex is dominated exactly once. In this paper, we investigate efficient multiple domination. There are several types of multiple domination defined in the literature: k-tuple domination, {k}-domination, and k-domination. We investigate ...

Department of Mathematics, National Taiwan University, Taipei, Taiwan Taida Institute for Mathematical Sciences, National Taiwan University, Taipei, Taiwan National Center for Theoretical Sciences, Taipei Office, Taiwan LaBRI UMR CNRS 5800, Univ. Bordeaux, Talence, France Department of Mathematics Education, Catholic University of Daegu, Kyongsan, Republic of Korea Department of Mathematics, Sh...

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The dom...