Distance domination and distance irredundance in graphs

Abstract

A set D ⊆V of vertices is said to be a (connected) distance, k-dominating set of G if the distance between each vertex υ ∈V - D and D is at most k (and D induces a connected graph in G). The minimum cardinality of a (connected) distance k-dominating set in G is the (connected) distance k-domination number of G, denoted byγ k(G) (γ kc(G), respectively). The set D is defined to be a total k-dominating set of G if every vertex in V is within distance k from some vertex of D other than itself. The minimum cardinality among all total k-dominating sets of G is called the total k-domination number of G and is denoted by γkt(G). For x ∈ X ⊆ V, if Nk[x] - Nk[X - χ] ≠ φ, the vertex x is said to be k-irredundant in X. A set X containing only k-irredundant vertices is called k-irredundant. The k-irredundance number of G, denoted by irk (G), is the minimum cardinality taken over all maximal k-irredundant sets of vertices of G. In this paper we establish lower bounds for the distance k-irredundance number of graphs and trees. More precisely, we prove that 5k+1/2 irk(G) ≥ γkc(G) + 2k for each connected graph G and (2k + 1)irk(T) ≥ γkc(T) + 2k ≥ |V| +2k- kn1(T) for each tree T = (V,E] with n1(T) leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann [9] and Cyman, Lernańska and Raczek [2] regarding γk. and the first generalizes a result of Favaron and Kratsch [4] regarding ir1. Furthermore, we shall show that γkc(G) ≥3k+1/2(G) - 2k for each connected graph G, thereby generalizing a result of Favaron and Kratsch [4] regarding k = 1.