An Algorithm to Find Two Distance Domination Parameters in a Graph

Let n > 1 be an integer and let G be a graph of order p. A set 9 of vertices of G is a total n-dominating set of G if every vertex of V(G) is within distance n from some vertex of .P’ other than itself. The minimum cardinal@ among all total n-dominating sets of G is called the total n-domination number and is denoted by “J:(G). A set S of vertices of G is n-independent if the distance (in G) between every pair of distinct vertices of S is at least n + 1. The minimum cardinality among all maximal n-independent sets of G is called the n-independence number of G and is denoted by in(G)_ In this paper, we present an algorithm for finding a total n-dominating set 9 and a maximal n-independent set S in a connected graph with at least p 2 2n + I vertices. It is shown that these sets 9 and S satisfy the inequality IS( + nlg\ 2n + 1 vertices, then i,(G) + n y:(G) < p.