The Partition Dimension of a Subdivision of a Complete Graph

Abstract

Abstract The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = { L 1, L 2, L 3, · · ·, 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 , L 1), d ( v , L 2), · · ·, 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.