If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number γc(G) of G. A graph G is called a perfect connected-dominant graph if γ(H) = γc(H) for each connected induced subgraph H of G. We prove that a graph is a perfect connected-dominant graph if and only i...

An asteroidal triple is a set of three independent vertices such that between any two of them there exists a path that avoids the neighl.)ourhood of the third. Graphs that do not. co~,tain an asteroidal triple are called asteroidal triple-free (AT-free) graphs. AT-free graphs strictly contain the well-known class of cocomparability graphs, and are not necessarily perfect.. We present efficient,...

Let G be a connected graph with vertex set V (G). A set S of vertices in G is called a weakly connected dominating set of G if (i) S is a dominating set of G and (ii) the graph obtained from G by removing all edges joining two vertices in V (G) \ S is connected. A weakly connected dominating set S of G is said to be minimum or a γw-set if |S| is minimum among all weakly connected dominating set...

Let G = (V, E) be a graph. Set D ⊆ V (G) is a total outerconnected dominating set of G if D is a total dominating set in G and G[V (G)−D] is connected. The total outer-connected domination number of G, denoted by γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γtc(T ) ≥ d 2n 3 e. Moreover, we constructively charact...

Given a graph, a connected dominating set is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected.A minimum connected dominating set is such a vertex subset with minimum cardinality. In this paper, we present a new one-step greedy approximation with performance ratio ln + 2 where is the maxi...

For a given graph G = (V,E), a set D ⊆ V (G) is said to be an outerconnected dominating set if D is dominating and the graph G−D is connected. The outer-connected domination number of a graph G, denoted by γ̃c(G), is the cardinality of a minimum outer-connected dominating set of G. We study several properties of outer-connected dominating sets and give some bounds on the outer-connected dominati...

To improve the efficiency of routing and broadcast and reducing energy consumption in the process of data transmission, calculating minimum connected dominating set is always used to construct virtual backbone network in wireless sensor networks. Calculating the minimum connected dominating set (MCDS) of plane graphs is a NPcomplete problem. In this paper, an algorithm leveraging 1hop neighborh...

Nodes of minimum connected dominating set (MCDS) form a virtual backbone in a wireless adhoc network. In this paper, a modified approach is presented to determine MCDS of an underlying graph of a Wireless Adhoc network. Simulation results for a variety of graphs indicate that the approach is efficient in determining the MCDS as compared to other existing techniques.

Let G = (V,E) be a graph. A set D ⊆ V is a total outer-connected dominating set of G if D is dominating and G[V −D] is connected. The total outer-connected domination number of G, denoted γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. It is known that if T is a tree of order n ≥ 2, then γtc(T ) ≥ 2n 3 . We will provide a constructive characterization for tre...

A vertex subset S in a graph G is a dominating set if every vertex not contained in S has a neighbor in S. A dominating set S is a connected dominating set if the subgraph G[S] induced by S is connected. A connected dominating set S is a minimal connected dominating set if no proper subset of S is also a connected dominating set. We prove that there exists a constant ǫ > 10 such that every grap...