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.