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 \(G\).