Nearly Perfect Sets in Graphs

In a graph G = (V; E), a set of vertices S is nearly perfect if every vertex in V \Gamma S is adjacent to at most one vertex in S. Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and efficient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S, the set S \Gamma fug is not nearly perfect. Similarly, a nearly perfect set S is 1-maximal if for every vertex u in V \Gamma S, S [ fug is not a nearly perfect set. Lastly, we define n p (G) to be the minimum cardinality of a 1-maximal nearly perfect set, and N p (G) to be the maximum cardinality of a 1-minimal nearly perfect set. In this paper we calculate these parameters for some classes of graphs. We show that the decision problem for n p (G) is NP-complete; we give a linear algorithm for determining n p (T ) for any tree T ; and we show that N p (G) can be calculated for any graph G in polynomial time. 1 Introduction Let G = (V; E) be a graph. We say..

On total restrained domination in graphs

summary:In this paper we initiate the study of total restrained domination in graphs. Let $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by $\gamma _r^t(G)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for $\gamma _r^t(G)$ are given in Section 2. Then the Nordhaus-Gaddum-type results for total restrained domination number are established in Section 3. Finally, we show that the decision problem for $\gamma _r^t(G)$ is NP-complete even for bipartite and chordal graphs in Section 4