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...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

We investigate a domination-like problem from the exact exponential algorithms viewpoint. The classical Dominating Set problem ranges among one of the most famous and studied NP -complete covering problems [6]. In particular, the trivial enumeration algorithm of runtime O∗(2n) 4 has been improved to O∗(1.4864n) in polynomial space, and O∗(1.4689n) with exponential space [9]. Many variants of th...

Recently, Zhang et al. [6] obtained some lower bounds of the signed domination number of a graph. In this paper, we obtain some new lower bounds of the signed domination number of a graph which are sharper than those of them.

A signed dominating function of a graph G with vertex set V is a function f : V → {−1, 1} such that for every vertex v in V the sum of the values of f at v and at every vertex u adjacent to v is at least 1. The weight of f is the sum of the values of f at every vertex of V . The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we study the...

 Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

A subset S of a vertex set of a graphG is a total (k, r)-dominating set if every vertex u ∈ V (G) is within distance k of at least r vertices in S. The minimum cardinality among all total (k, r)-dominating sets ofG is called the total (k, r)domination number of G, denoted by γ (k,r)(G). We previously gave an upper bound on γ t (2,r)(G(n, p)) in random graphs with non-fixed p ∈ (0, 1). In this p...