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 Pi = {L-1, L-2, L-3, ..., L-k} be a k-partition of V(G). The representation r(v|.) of a vertex v with respect to Pi is the vector (d(v, L-1), d(v, L-2), ..., d(v, L-k)). The partition Pi is called a resolving partition of G if r(w vertical bar Pi) not equal r(v vertical bar Pi) for all distinct w, v epsilon 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 K-n with n >= 9. Additionally for n epsilon [2, 8], we obtain the exact values of the partition dimensions. (C) 2015 The Authors. Published by Elsevier B.V.