Partition dimension of certain classes of series parallel graphs

The partition dimension problem is to decompose the vertex set of a graph into minimum number of vertex disjoint sets such that the ordered distances between every vertex of the graph and the disjoint sets of that decomposition have different values in the representation. This problem has emerging applications in areas such as structure-activity issues in drug design, navigation of robots, pattern recognition and image processing. In the present study, we provide an upper bound for the partition dimension of parallel composition of any graphs and derive an exact result for parallel composition of paths with different lengths. On the other side, we find the partition dimension of the series composition of cycles and complete graphs with different sizes and an upper bound is derived for series composition of any graphs. Further, this problem is applied to a graph which has been generated by parallel and series composition together on paths.