Document Type : Research Articles


1 University of Mazandaran

2 Department of Mathematics, University of Tafresh



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.