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 script D sign 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 script D sign other than itself. The minimum cardinality among all total n-dominating sets of G is called the total n-domination number and is denoted by γ^t_n(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 i_n(G). In this paper, we present an algorithm for finding a total n-dominating set script D sign and a maximal n-independent set S in a connected graph with at least p≥2n+1 vertices. It is shown that these sets script D sign and S satisfy the inequality |S| + n|script D sign|≤p. Using this result, we conclude that if G is a connected graph on p≥2n+1 vertices, then i_n(G) + n · γ^t_n(G)≤p.