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.