A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) − k(G), and if G is a graph of order n with at most one cycle, then γ2(G) > (n+ l...

In this paper, we continue the study of power domination in graphs (see SIAM J. Discrete Math. 15 (2002), 519–529; SIAM J. Discrete Math. 22 (2008), 554–567; SIAM J. Discrete Math. 23 (2009), 1382–1399). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to b...

For a simple graph G, the independent domination number i(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds for the independent domination number of K1,k+1-free graphs, as functions of the order, size and k. Also we present a lower bound for the size of connected graphs with given order and value of independent domination ...

In this paper we introduce the concept of strong total domination in fuzzy graphs. We determine the strong total domination number for several classes of fuzzy graphs. A lower bound and an upper bound for the strong total domination number in terms of strong domination number is obtained. Strong total domination in fuzzy trees is studied. A necessary and sufficient condition for the set of fuzz...

An exact lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: (×i=1Kni) ≥ t + 1, t ≥ 3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of tw...

The total domination number of G denoted by γt(G) is the minimum cardinality of a total dominating set of G. A graph G is total domination vertex critical or just γt-critical, if for any vertex v of G that is not adjacent to a vertex of degree one, γt(G − v) < γt(G). If G is γt-critical and γt(G) = k, then G is k-γt-critical. Haynes et al [The diameter of total domination vertex critical graphs...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = ∑ v∈V f(v). The k-distance Roman domination number ...

For any integer k ≥ 1, a signed (total) k-dominating function is a function f : V (G) → {−1, 1} satisfying w∈N [v] f(w) ≥ k ( P w∈N(v) f(w) ≥ k) for every v ∈ V (G), where N(v) = {u ∈ V (G)|uv ∈ E(G)} and N [v] = N(v)∪{v}. The minimum of the values ofv∈V (G) f(v), taken over all signed (total) k-dominating functions f, is called the signed (total) k-domination number and is denoted by γkS(G) (γ...

Henning, A.M. and H.C. Swart, Bounds relating generalized domination parameters, Discrete Mathematics 120 (1993) 933105. The domination number r(G) and the total domination number y,(G) of a graph G are generalized to the K,-domination number 7x,(G) and the total K,-domination number y;_(G) for n>2, where y(G)=y,,(G) and Y~(G)=~~~(G), K,-connectivity is defined and, for every integer n >2, the ...

The domination number of a graph G = (V,E) is the minimum size of a dominating set U ⊆ V , which satisfies that every vertex in V \U is adjacent to at least one vertex in U . The notion of a problem kernel refers to a polynomial time algorithm that achieves some provable reduction of the input size. Given a graph G whose domination number is k, the objective is to design a polynomial time algor...