Partition dimension of generalized Peterson and Harary graphs

The distance of a connected, simple graph \(\mathbb{P}\) is denoted by \(d({\alpha}_1,{\alpha}_2),\) which is the length of a shortest path between the vertices \({\alpha}_1,{\alpha}_2\in V(\mathbb{P}),\) where \(V(\mathbb{P})\) is the vertex set of \(\mathbb{P}.\) The \(l\)-ordered partition of \(V(\mathbb{P})\) is \(K=\{K_1,K_2,\dots,K_l\}.\) A vertex \({\alpha}\in V(\mathbb{P}),\) and \(r({\alpha}|K)=\{d({\alpha},K_1),d({\alpha},K_2),\dots,d({\alpha},K_l)\}\) be a \(l\)-tuple distances, where \(r({\alpha}|K)\) is the representation of a vertex \({\alpha}\) with respect to set \(K.\) If \(r({\alpha}|K)\) of \({\alpha}\) is unique, for every pair of vertices, then \(K\) is the resolving partition set of \(V(\mathbb{P}).\) The minimum number \(l\) in the resolving partition set \(K\) is known as partition dimension (\(pd(\mathbb{P})\)). In this paper, we studied the generalized families of Peterson graph, \(P_{{\lambda},{\chi}}\) and proved that these families have bounded partition dimension.