The Partition Dimension of Corona Product Graphs

Abstract

Given a set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\) of a connected graph \(G\), the metric representation of a vertex \(v\) of \(G\) with respect to \(S\) is the vector \(r(v|S) = (d(v, v_1), d(v, v_2), \ldots, d(v, v_k))\), where \(d(v, v_i)\), \(i \in \{1, \ldots, k\}\), denotes the distance between \(v\) and \(v_i\). \(S\) is a resolving set of \(G\) if for every pair of distinct vertices \(u, v\) of \(G\), \(r(u|S) \neq r(v|S)\). The metric dimension \(\dim(G)\) of \(G\) is the minimum cardinality of any resolving set of \(G\). Given an ordered partition \(\Pi = \{P_1, P_2, \ldots, P_t\}\) of vertices of a connected graph \(G\), the partition representation of a vertex \(v\) of \(G\), with respect to the partition \(\Pi\), is the vector \(r(v|\Pi) = (d(v, P_1), d(v, P_2), \ldots, d(v, P_t))\), where \(d(v, P_i)\), \(1 \leq i \leq t\), represents the distance between the vertex \(v\) and the set \(P_i\), that is \(d(v, P_i) = \min_{u \in P_i} \{d(v, u)\}\). \(\Pi\) is a resolving partition for \(G\) if for every pair of distinct vertices \(u, v\) of \(G\), \(r(u|\Pi) \neq r(v|\Pi)\). The partition dimension \(\mathrm{pd}(G)\) of \(G\) is the minimum number of sets in any resolving partition for \(G\). Let \(G\) and \(H\) be two graphs of order \(n\) and \(m\), respectively. The corona product \(G \odot H\) is defined as the graph obtained from \(G\) and \(H\) by taking one copy of \(G\) and \(n\) copies of \(H\) and then joining, by an edge, all the vertices from the \(i\)-th copy of \(H\) with the \(i\)-th vertex of \(G\). Here, we study the relationship between \(\mathrm{pd}(G \odot H)\) and several parameters of the graphs \(G \odot H\), \(G\), and \(H\), including \(\dim(G \odot H)\), \(\mathrm{pd}(G)\), and \(\mathrm{pd}(H)\).