For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination...

‎For any integer $kgeq 1$‎, ‎a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-‎tuple total dominating set of $G$ if any vertex‎ ‎of $G$ is adjacent to at least $k$ vertices in $S$‎, ‎and any vertex‎ ‎of $V-S$ is adjacent to at least $k$ vertices in $V-S$‎. ‎The minimum number of vertices of such a set‎ ‎in $G$ we call the $k$-tuple total restrained domination number of $G$‎. ‎The maximum num...

‎for any integer $kgeq 1$‎, ‎a set $s$ of vertices in a graph $g=(v,e)$ is a $k$-‎tuple total dominating set of $g$ if any vertex‎ ‎of $g$ is adjacent to at least $k$ vertices in $s$‎, ‎and any vertex‎ ‎of $v-s$ is adjacent to at least $k$ vertices in $v-s$‎. ‎the minimum number of vertices of such a set‎ ‎in $g$ we call the $k$-tuple total restrained domination number of $g$‎. ‎the maximum num...

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any in...

In this paper, a necessary and sufficient condition for the existence of an efficient 2-dominating set in a class of circulant graphs has been obtained and for those circulant graphs, an upper bound for the 2domination number is also obtained. For the circulant graphs Cir(n,A), where A = {1, 2, . . . , x, n − 1, n − 2, . . . , n − x} and x ≤ bn−1 2 c, the perfect 2-tuple total domination number...

a r t i c l e i n f o a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = (V , E) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k. In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple ...

Given a simple graph G = (V, E) and a fixed positive integer k. In a graph G, a vertex is said to dominate itself and all of its neighbors. A set D ⊆ V is called a k-tuple dominating set if every vertex in V is dominated by at least k vertices of D. The k-tuple domination problem is to find a minimum cardinality k-tuple dominating set. This problem is NP-complete for general graphs. In this pap...

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...