On signed distance-k-domination in graphs

Abstract

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 Nk(v), is the set Nk(v) = {u: u ≠ v and d(u, v) ⩽ k}. Nk[v] = Nk(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 \(v \in V, f(N_k [v]) = \sum\limits_{u \in N_k [v]} {f(u) \geqslant 1} \). The signed distance-k-domination number, denoted by γk,s(G), is the minimum weight of a signed distance-k-dominating function on G. The values of γ2,s(G) are found for graphs with small diameter, paths, circuits. At the end it is proved that γ2,s(T) is not bounded from below in general for any tree T.