The partition dimension of the vertex amalgamation of some cycles

Let [Formula: see text] be a connected, finite, simple, and undirected graph. The distance between two vertices [Formula: see text] , denoted by [Formula: see text] , is the shortest length of [Formula: see text]-path in G. The distance between a vertex [Formula: see text] is defined as [Formula: see text] where [Formula: see text] , denoted by [Formula: see text]. For an ordered partition [Formula: see text] of the vertices of a graph G, the partition representation of a vertex [Formula: see text] with respect to Π is defined as the k-vektor [Formula: see text]. The partition set Π is called a resolving partition of G, if [Formula: see text] , for all [Formula: see text] , [Formula: see text]. The partition dimension of G is the minimum number of sets in any resolving partition of G. In this paper we study the partition dimension of the vertex amalgamation of some cycles. Specifically, we present the vertex amalgamation of m copies of the cycle [Formula: see text] at a fixed vertex [Formula: see text] , for [Formula: see text] and [Formula: see text] , [Formula: see text].