Distance−$k$ unique isolate perfect domination on graphs

Abstract

A domination set $D$ of a graph $G$ is perfect if each vertex of $V(G)-D$ is dominated by exactly one vertex in $D$. A dominating set $D$ is called $k$-perfect if for every $u \in V-D$ there exists a unique vertex $w \in D$ such that $d(u, D)=d(u, w) \leq k$. For an integer $k \geq 1, D \subseteq V(G)$ is a distance $k$-dominating set of $G$, if every vertex in $V(G)-D$ is within the distance $k$ from some vertex $v \in D$. That is, $N_k[D]=V(G)$. A distance $-k$ perfect dominating set $D$ of $G$ is said to be a distance $-k$ UIPDS of $G$ if $\langle D\rangle$ has exactly one isolated vertex and $D$ is $k$-perfect. This paper includes some properties of distance- $k$ UIPDS and gives the distance $-k$ UIPD number of paths, cycles, complete a-partite graphs, disconnected graphs and some directed graphs.