On signed distance-k-domination in graphs

The signed distance-k-domination number of a graph is a certain variant of the signed domination number. If v is a vertex of a graph G, the open k-neighborhood of v, denoted by N _k (v), is the set N _k (v) = {u: u ≠ v and d(u, v) ⩽ k}. N _k v] = N _k (v) ⋃ {v} is the closed k-neighborhood of v. A function f: V → {−1, 1} is a signed distance-k-dominating function of G, if for every vertex . The signed distance-k-domination number, denoted by γ _k,s (G), is the minimum weight of a signed distance-k-dominating function on G. The values of γ _2,s (G) are found for graphs with small diameter, paths, circuits. At the end it is proved that γ _2,s (T) is not bounded from below in general for any tree T.