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 minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph...

A 2-directional orthogonal ray graph is an intersection graph of rightward rays (half-lines) and downward rays in the plane. We show a dynamic programming algorithm that solves the weighted dominating set problem in O(n3) time for 2-directional orthogonal ray graphs, where n is the number of vertices of a graph. key words: Boolean-width, dominating set, dynamic programming, twodirectional ortho...

In this paper we introduce the common minimal equitable and vertex minimal equitable dominating graph and we get characterize the common minimal equitable and vertex minimal equitable dominating graph which are either connected or complete, some new results of these graphs are obtained. Mathematics Subject Classification: 05C70

A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S . If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S , then it is said to be locating-dominating. Locating, metric-locating-dominating and locatingdom...

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...

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. The minimum cardinality taken over all, the minimal double dominating set which is called Fuzzy Double Domination Number and which is denoted as ) (G fdd  . A set V S  is called a Triple dominating set of a graph G if every ...

We give the bipartite theory of vertex-edge weak dominating set and edge-vertex strong dominating set of a graph. Mathematics Subject Classification: 05C69

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

For any graph G = (V, E), D  V is a global dominating set if D dominates both G and its complement G . The global domination number g(G) of a graph G is the fewest number of vertices required of a global dominating set. In general, max{(G), (G )} ≤ g(G) ≤ (G)+(G ), where (G) and (G ) are the respective domination numbers of G and G . We show, when G is a planar graph, that g(G) ≤ max{...

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most (n−2√n )/4+O(log n) edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.