Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. We define the restrained bondage number br(G) of a nonempty graph G to be the minimum cardinality among all sets of edges E′ ⊆...

Given a semigraph, we can construct graphs Sa, Sca, Se and S1e. In the same pattern, we construct bipartite graphs CA(S), A(S), VE(S), CA(S) and A(S). We find the equality of domination parameters in the bipartite graphs constructed with the domination and total domination parameters of the graphs Sa and Sca. We introduce the domination and independence parameters for the bipartite semigraph. W...

We consider a dynamic domination problem for graphs in which an infinite sequence of attacks occur at vertices with guards and the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack and the resulting guard movements, the vertices containing guards form ...

We consider the problem of nding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We rst give an approximation algorithm for at most cubic graphs, that achieves ratio 2, based on greedy and local search techniques. We then propose an heuristic based on an iterative application of the greedy technique on graphs of lower and lower degree. When the graph i...

Let ν be some graph parameter and let G be a class of graphs for which ν can be computed in polynomial time. In this situation it is often possible to devise a strategy to decide in polynomial time whether ν has a unique realization for some graph in G. We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for thre...

Let G=(V,E) be a graph. A subset S of V is a dominating set of G if every vertex in V \ S is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for each v V \ S there exists u S such that v is adjacent to u and S1=(S-{u}) {v} is a dominating set. In this paper we introduce the concept of secure irredundant set and obtain an inequality chain of four parameters.

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set ofG if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investi...

We give a constructive characterization of trees that have a maximum independent set and a minimum dominating set which are disjoint and show that the corresponding decision problem is NP-complete for general graphs.

The six basic parameters relating to domination, independence and irredundance satisfy a chain of inequalities given by ir ≤ γ ≤ i ≤ β0 ≤ Γ ≤ IR where ir, IR are the irredundance and upper irredundance numbers, γ,Γ are the domination and upper domination numbers and i, β0 are the independent domination number and independence number respectively. In this paper, we introduce the concept of indep...

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. A vertex that is contained in some minimum total dominating set of a graph G is a good vertex, otherwise it is a bad vertex. We determine for which triples (x, y, z) there exists a conne...