Family of Graphs with Partition Dimension Three

The characterization of all connected graphs of order <em>n</em> ≥3 with partition dimension 2, <em>n</em>−1 or <em>n</em> has been completely done. Additionally, all connected graphs of order <em>n</em>≥9 with partition dimension <em>n</em>−2 and graphs of order <em>n</em>≥11 with partition dimension <em>n</em>−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.