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.