Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P7-free chordal gr...

For many fixed-parameter problems that are trivially soluable in polynomial time, such as (k-)DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as (k-)FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed k the problem is soluable in time bounded by a polynomial of degree c, where c is a c...

For a positive integer k, a set of vertices S in a graph G is said to be a kdominating set if each vertex x in V (G)−S has at least k neighbors in S. The order of a smallest k-dominating set of G is called the k-domination number of G and is denoted by γk(G). In Blidia, Chellali and Favaron [Australas. J. Combin. 33 (2005), 317–327], they proved that a tree T satisfies α(T ) ≤ γ2(T ) ≤ 3 2 α(T ...

The complete cototal domination set is said to be irredundant dominating if for each u ∈ S, NG [S − {u}] ≠ [S]. minimum cardinality taken over all an called number and denoted by γircc(G). Here a new parameter was introduced the study of bounds γircc(G) initiated.

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...

The idea of virtual backbone routing for ad hoc wireless networks is to operate routing protocols over a virtual backbone. One purpose of virtual backbone routing is to alleviate the serious broadcast storm problem suffered by many exiting on-demand routing protocols for route detection. Thus constructing a virtual backbone is very important. In our study, the virtual backbone is approximated b...

Connected Dominating Set is widely used as virtual backbone in wireless Ad-hoc and sensor networks to improve the performance of transmission and routing protocols. Based on special characteristics of Ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacen...

The concept of triple connected graphs with real life application was introduced in [7] by considering the existence of a path containing any three vertices of a graph G. In this paper, we introduce a new domination parameter, called Smarandachely triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be Smarandachely triple connected dominating set, i...

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in ...