Minimal Restrained Domination Algorithms on Trees Using Dynamic Programming

In this paper we study a special case of graph domination, namely minimal restrained dominating sets on trees. A set S ? V is a dominating set if for every vertex u ? V-S, there exists v ? S such that uv ? E. A set S ? V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and another vertex in V-S. A restrained dominating set S is a minimal restrained dominating set if no proper subset of S is also a restrained dominating set. We give a dynamic programming style algorithm for generating largest minimal restrained dominating sets for trees and show that the decision problem for minimal restrained dominating sets is NP-complete for general graphs. We also consider independent restrained domination on trees and its associated decision problem for general graphs

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

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 $n/2$