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Abstract
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For an integer $k\geq1$, a $k$-distance enclaveless number (or $k$-distance $B$-differential) of a connected graph $G=(V,E)$ is $\Psi^k(G)=max\{|(V-X)\cap N_{k,G}(X)|:X\subseteq V\}$. 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, $\Psi^k(G\times H)\leq mn-n-m+\Psi^k(G)+\Psi^k(H)+1$, where $G\times H$ 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 $n\geq k+1$ vertices and with $n_1$ leaves satisfies inequality $\Psi^k(T)\leq\frac{k(2n-2+n_1)}{2k+1}$ and we characterize the extremal trees.
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