For a positive integer k, a k-subdominating function of a graph G=(V; E) is a function f :V →{−1; 1} such that ∑u∈NG [v] f(u)¿ 1 for at least k vertices v of G. The ksubdomination number of G, denoted by ks(G), is the minimum of ∑ v∈V f(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We al...

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1,1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed do...

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...

Let G = (V, E) be a simple graph with vertex set V and edge set E. A function f from V to a set {-1, 1} is said to be a nonnegative signed dominating function (NNSDF) if the sum of its function values over any closed neighborhood is at least zero. The weight of f is the sum of function values of vertices in V. The nonnegative signed domination number for a graph G equals the minimum weight of a...

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...

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

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.