Distance Domination and Distance Irredundance in Graphs

A set D ⊆ V of vertices is said to be a (connected) distance k-dominating set of G if the distance between each vertex u ∈ 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 by γk(G) (γ c k (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 γ t k (G). For x ∈ X ⊆ V , if N[x] − N[X − x] 6= ∅, 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) ≥ γ c k (G) + 2k for each connected graph G and (2k + 1)irk(T ) ≥ γ c k (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, Lemańska and Raczek [2] regarding γk and the first generalizes a result of Favaron and Kratsch [4] regarding ir1. Furthermore, we shall show that γ c k (G) ≤ 3k+1 2 γ k (G) − 2k for each connected graph G, thereby generalizing a result of Favaron and Kratsch [4] regarding k = 1.