$k$-distance enclaveless number of a graph

For an integer $kgeq1$, a $k$-distance enclaveless number (or $k$-distance $B$-differential) of a connected graph $G=(V,E)$ is $Psi^k(G)=max{|(V-X)cap N_{k,G}(X)|:Xsubseteq V}$. 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, $Psi^k(Gtimes H)leq mn-n-m+Psi^k(G)+Psi^k(H)+1$, where $Gtimes H$ denotes the direct product of $G$ and $H$. In the end of this paper, we show that the $k$-distance enclaveless number $Psi^k(T)$ of a tree $T$ on $ngeq k+1$ vertices and with $n_1$ leaves satisfies inequality $Psi^k(T)leqfrac{k(2n-2+n_1)}{2k+1}$ and we characterize the extremal trees.