Bounds on several versions of restrained domination number

We investigate several versions of restraineddomination numbers and present new bounds on these parameters. We generalize theconcept of restrained domination and improve some well-known bounds in the literature.In particular, for a graph $G$ of order $n$ and minimum degree $\delta\geq 3$, we prove thatthe restrained double domination number of $G$ is at most $n-\delta+1$. In addition,for a connected cubic graph $G$ of order $n$ we show thatthe total restrained domination number of $G$ is at least $n/3$ andthe restrained double domination number of $G$ is at least $n/2$

Characterizations in Domination Theory

Let G = (V,E) be a graph. A set R is a restrained dominating set (total restrained dominating set, resp.) if every vertex in V − R (V) is adjacent to a vertex in R and (every vertex in V −R) to a vertex in V −R. The restrained domination number of G (total restrained domination number of G), denoted by gamma_r(G) (gamma_tr(G)), is the smallest cardinality of a restrained dominating set (total restrained dominating set) of G. If T is a tree of order n, then gamma_r(T) is greater than or equal to (n+2)/3. We show that gamma_tr(T) is greater than or equal to (n+2)/2. Moreover, we show that if n is congruent to 0 mod 4, then gamma_tr(T) is greater than or equal to (n+2)/2 + 1. We then constructively characterize the extremal trees achieving these lower bounds. Finally, if G is a graph of order n greater than or equal to 2, such that both G and G\u27 are not isomorphic to P_3, then gamma_r(G) + gamma_r(G\u27) is greater than or equal to 4 and less than or equal to n +2. We provide a similar result for total restrained domination and characterize the extremal graphs G of order n achieving these bounds

On Roman, Global and Restrained Domination in Graphs

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed. © 2010 Springer