The domination number γ(G) of the fuzzy graph G is the minimum cardinality taken over all minimal dominating sets of G. The independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of G. The irredundant number ir(G) is the minimum cardinality taken over all maximal irredundant sets of G. In this paper we prove the result that relate the parameters ir...

Let G = (V (G), E(G)) be a simple graph, and let k be a positive integer. A subset D of V (G) is a k-dominating set if every vertex of V (G) − D is dominated at least k times by D. The k-domination number γk(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ2(T ) ≥ γ1(T ) + 1 and characterized extremal trees attaining this bound....

A triangle-free graph is maximal if the addition of any edge produces a triangle. A set S of vertices in a graph G is called an independent dominating set if S is both an independent and a dominating set of G. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. In this paper, we show that i(G) ≤ δ(G) ≤ n 2 for maximal triangle-free graph...

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number and the total 2-tuple domination number of the factors. Using these relationships some exact total domination numbers are obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direc...

In this note the split domination number of the Cartesian product of two paths is considered. Our results are related to [2] where the domination number of Pm¤Pn was studied. The split domination number of P2¤Pn is calculated, and we give good estimates for the split domination number of Pm¤Pn expressed in terms of its domination number.

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, the sums and products of ψ(G1) and ψ(G2) are examined where G1 ⊕ G2 = K(s, s), and ψ is the independence, domination, or independent domination number, inter alia. In particular, it is shown that the maximum valu...

Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any integers k1, k2, k3, k4, k5 there exists a cubic graph G satisfying the following conditions: γ(G)−ir(G) ≥ k1, i(G)−...

We prove a new upper bound on the independent domination number of graphs in terms of the number of vertices and the minimum degree. This bound is slightly better than that by J. Haviland 3] and settles Case = 2 of the corresponding conjecture by O. Favaron 2].

A dominating set D of a graph G is a subset of V (G) such that for every vertex v ∈ V (G), either in v ∈ D or there exists a vertex u ∈ D that is adjacent to v. We are interested in finding dominating sets of small cardinality. A dominating set I of a graph G is said to be independent if no two vertices of I are connected by an edge of G. The size of a smallest independent dominating set of a g...

We settle two conjectures on domination-search, a game proposed by Fomin et.al. [1], one in affirmative and the other in negative. The two results presented here are (1) domination search number can be greater than domination-target number, (2) domination search number for asteroidal-triple-free graphs is at most 2.