K-distance enclaveless number of a graph

For an integer k ≥ 1, a k-distance enclaveless number (or k-distance B-differential) of a connected graph G = (V,E) is Ψk(G) = max{|(V − X) ∩ Nk,G(X)| : X ⊆ V }. In this paper, we establish upper bounds on the k-distance enclaveless number of a graph in terms of its diameter, radius and girth. Also, we prove that for connected graphs G and H with orders n and m respectively, Ψk(G × H) ≤ mn − n − m + Ψk(G) + Ψk(H) + 1, where G × H denotes the direct product of G and H. In the end of this paper, we show that the k-distance enclaveless number Ψk(T) of a tree T on n ≥ k + 1 vertices and with n1 leaves satisfies inequality Ψk(T) ≤ k(2n−2+n1) 2k+1 and we characterize the extremal trees.