Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

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...

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 ...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D)→ {−1,1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f (x) ≥ k for each v∈V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set { f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property that ∑...

For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for...

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.