For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid...

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...

This paper contains a number of estimations of the split domination number and the maximal domination number of a graph with a deleted subset of edges which induces a complete subgraph Kp. We discuss noncomplete graphs having or not having hanging vertices. In particular, for p = 2 the edge deleted graphs are considered. The motivation of these problems comes from [2] and [6], where the authors...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. 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 . we define a restrained roman dominating function on a graph g = (v,e) to be ...

An important technique in large cardinal set theory is that of extending an elementary embedding j : M → N between inner models to an elementary embedding j∗ : M [G] → N [G∗] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold, the generic G∗ is simply...

The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum car...

A set D ⊆ V (G) of a graph G = (V,E) is a liar’s dominating set if (1) for all v ∈ V (G) |N [v] ∩ D| ≥ 2 and (2) for every pair u, v ∈ V (G) of distinct vertices, |N [u] ∪ N [v] ∩ D| ≥ 3. In this paper, we consider the liar’s domination number of some middle graphs. Every triple dominating set is a liar’s dominating set and every liar’s dominating set must be a double dominating set. So, the li...

A total dominating set of a graph G = (V,E) is a subset D ⊆ V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NPcomplete on planar graphs and W [2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [FOCS 2009], it follows that there exists a linear kernel for Total Dominatin...