Metric Dimension of Graph Join P2 and Pt - Loeky Haryanto; Nurdin Nurdin; Hasmawati Hasmawati (IndoMS, 2019)
The following metric dimension of join two paths P_2 + P_t is determined as follows. For every k = 1, 2, 3, ... and t = 2 + 5k or t = 3 + 5k, the dimension of P_2 + P_t is 2 + 2k whereas for t = 4 + 5k, t = 5(k+1) or t = 1 + 5(k+1), the dimension is 3 + 2k. In case t ≥ 7, the dimension is determined by a chosen (maximal) ordered basis for P_2 + P_t in which the integers 1, 2 are the two consecutive vertices of P_2 and the next integers 3, 4, ..., t + 2 are the t consecutive vertices of P_t. If t ≥ 10, the ordered binary string contains repeated substrings of length 5. For t < 7, the dimension is easily found using a computer search, or even just using hand computations.