The partition dimension of the vertex amalgamation of some cycles

Abstract

Let G=(V(G),E(G)) be a connected, finite, simple, and undirected graph. The distance between two vertices u,w∈V(G), denoted by d(u,w), is the shortest length of (u,w)-path in G. The distance between a vertex v∈V(G) is defined as min⁡{d(v,x):x∈S} where S⊂V(G), denoted by d(v,S). For an ordered partition Π={S1,S2,…,Sk} of the vertices of a graph G, the partition representation of a vertex v∈V(G) with respect to Π is defined as the k-vektorr(v|Π)=(d(v,S1),d(v,S1),…,d(v,S1). The partition set Π is called a resolving partition of G, if r(u|Π)≠r(v|Π), for all u≠v, u,v∈V(G). 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 Cn at a fixed vertex v∈V(Cn), for n≥6 and k2−3k+42≤m≤k2−k2, k≥3.