The Partition Dimension of a Subdivision of a Complete Graph

The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = {L1, L2, L3, · · ·, Lk } be a k-partition of V(G). The representation r(v|Π) of a vertex v with respect to Π is the vector (d(v, L1), d(v, L2), · · ·, d(v, Lk)). The partition Π is called a resolving partition of G if r(w|Π) ≠ r(v|Π) for all distinct w, v ∈ V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G.This paper considers in finding partition dimensions of graphs obtained from a subdivision operation. In particular, we derive an upper bound of partition dimension of a subdivision of a complete graph Kn with n ≥ 9. Additionally for n ∈ [2,8], we obtain the exact values of the partition dimensions.