Family of Graphs with Partition Dimension Three - Haryeni, Debi Oktia; Baskoro, Edy Tri; Saputro, Suhadi Wido (Indonesian Journal of Combinatorics, 2024)
The characterization of all connected graphs of order n ≥3 with partition dimension 2, n−1 or n has been completely done. Additionally, all connected graphs of order n≥9 with partition dimension n−2 and graphs of order n≥11 with partition dimension n−3 have been characterized as well. However, the characterization of all connected graphs with partition dimension 3 is an open problem. In this paper, we construct many families of disconnected as well as connected graphs with partition dimension 3 by generalizing the concept of the partition dimension so that it can be applied to disconnected graphs.