A dominating set for a graph G is a subset D of V such that every vertex not in D is adjacent to at least one member of D. This paper deals with some of the graphs having unique minimum dominating sets. We also find a unique minimum dominating sets for block graphs and maximum graphs.

Eternal domination is a problem that asks the following question: Can we eternally defend graph ? The principle to against an attacked vertex, changes every turn, by moving guards along edges of graph. In classical version game (Burger et al., 2004), only one guard can move at time, but in m-eternal (Goddard 2005), any number single turn. This led introduction two parameters: eternal and number...

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) Dominating ($f$-BDS). Let $G=(V,E)$ be a graph with color assignment function $\gamma \colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ from each ...

This paper focuses on the average order of dominating sets a graph. We find extremal graphs for maximum and minimum value over all n vertices, while trees we prove that star minimizes sets. in without isolated vertices is at most 3n/4, but provide evidence actual upper bound 2n/3. Finally, show normalized average, dense [1/2,1], tends to 12 almost graphs.

Researchers propose Connected Dominating Set (CDS) of graphs in which each node in the cluster wireless network cover via dominating neighbors, define many dominating sets such as strongly connected dominating sets and weakly connected dominating sets etc. In this paper, we extend the dominative capacity of nodes such that each node dominates not only itself and all its adjacent neighbors compl...