On the metric dimension of the reduced power graph of a finite group
The problem of finding the metric dimension of a given graph is hard, determining whether the metric dimension is bounded by a given value. To understand the definition of the metric dimension of a given graph \(\Gamma\), we start as follows. Let \(W=\{v_1, v_2, \dots, v_k\} \subseteq V(\Gamma)\) and let \(v\) be any vertex of \(\Gamma\). The representation of \(v\) with respect to \(W\) is the \(k\)-vector \(r(v|W) = (d(v, v_1), d(v, v_2),\) \( \cdots, d(v, v_k))\). If distinct vertices have distinct representations with respect to \(W\), then \(W\) is called a resolving set for \(\Gamma\). A basis of \(\Gamma\) is a minimum resolving set for \(\Gamma\) and the cardinality of a basis of \(\Gamma\) is called the metric dimension of \(\Gamma\) and denoted by \(\dim(\Gamma)\). As an example, \textit{S. M. S. Khasraw} [``On intersection graph of dihedral group'', J. Math. Comput. Sci. 6. 6714--6728 (2021; \url{doi:10.28919/jmcs/6284})] found the metric dimension of the intersection graph of the dihedral group. In this paper, the authors considered the reduced power graph \(\mathcal{P}_R(G)\) of a finite group \(G\), which is a simple graph with the vertex set \(G\) and two distinct vertices \(x\) and \(y\) are adjacent if \(\langle x \rangle \subset \langle y \rangle\) or \(\langle y \rangle \subset \langle x \rangle\). They found upper and lower bounds of such graphs. In addition, the metric dimension of the reduced power graph of a \(\mathcal{P}\)-group, a cyclic group, a dihedral group, and a generalized quaternion group is computed.
scientific article; zbMATH DE number 7481699
scientific article