We develop the theory of domination by stable types and stable weight in an arbitrary theory.

We find bounds for the domination number of a tournament and investigate the sharpness of these bounds. We also find the domination number of a random tournament.

For graphs G and H , a set S ⊆ V (G) is an H -forming set of G if for every v∈V (G) − S, there exists a subset R ⊆ S, where |R|= |V (H)| − 1, such that the subgraph induced by R∪{v} contains H as a subgraph (not necessarily induced). The minimum cardinality of an H -forming set of G is the H -forming number {H}(G). The H -forming number of G is a generalization of the domination number (G) beca...

A global defensive alliance in a graph G = (V,E) is a dominating set S satisfying the condition that for every vertex v ∈ S, |N [v] ∩ S| ≥ |N(v) ∩ (V − S)|. In this note, a new upper bound on the global defensive alliance number of a tree is given in terms of its order and the number of support vertices. Moreover, we characterize trees attaining this upper bound.

Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= ...

We show that the total domination number of a graph G whose complement , G c , does not contain K3;3 is at most (G c), except for complements of complete graphs, and graphs belonging to a certain family which is characterized. In the case where G c does not contain K4;4 we show that there are four exceptional families of graphs, and determine the total domination number of the graphs in each one.

A function f de1ned on the vertices of a graph G = (V; E); f :V → {−1; 0; 1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V ) = ∑ v∈V f(v). The minus domination number of a graph G, denoted by −(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bo...

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...