On the Connected Partition Dimension of Unicyclic Graphs

Abstract

Let \( G \) be a connected graph. For a vertex \( v \in V(G) \) and an ordered \( k \)-partition \( \Pi = (S_1, S_2, \dots, S_k) \) of \( V(G) \), the representation of \( v \) with respect to \( \Pi \) is the \( k \)-vector \( r(v|\Pi) = (d(v, S_1), d(v, S_2), \dots, d(v, S_k) ) \) where \( d(v, S_i) = \min_{w \in S_i} d(x, w) \) (\( 1 \leq i \leq k \)). The \( k \)-partition \( \Pi \) is said to be resolving if the \( k \)-vectors \( r(v|\Pi) \), \( v \in V(G) \), are distinct. The minimum \( k \) for which there is a resolving \( k \)-partition of \( V(G) \) is called the partition dimension of \( G \), denoted by \( pd(G) \). A resolving \( k \)-partition \( \Pi = \{ S_1, S_2, \dots, S_k \} \) of \( V(G) \) is said to be connected if each subgraph \( \langle S_i \rangle \) induced by \( S_i \) (\( 1 \leq i \leq k \)) is connected in \( G \). The minimum \( k \) for which there is a connected resolving \( k \)-partition of \( V(G) \) is called the connected partition dimension of \( G \), denoted by \( cpd(G) \). In this paper, the connected partition dimension of the unicyclic graphs is calculated and bounds are proposed.