k-perfect geodominating sets in graphs

Abstract

A perfect geodominating set in a graph $G$ is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair of vertices of $S$. A $k$-perfect geodominating set is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair $x$, $y$ of vertices of $S$ with $d(x, y) = k$. We study perfect and $k$-perfect geodomination numbers of a graph $G$.