The spanning-tree upper-bound conjecture for partition dimension of unicyclic graphs

Let $G$ be a unicyclic graph and let $T$ be a spanning tree of $G$. The spanning-tree upper-bound conjecture. If $T$ is a spanning tree of a unicyclic graph $G$, then

$$\operatorname{pd}(G)\leq \operatorname{pd}(T)+1.$$

This conjecture proposes an upper bound for the partition dimension of a unicyclic graph in terms of that of a spanning tree, analogous to the known estimate for metric dimension. The preceding discussion notes that the bound is attained for certain unicyclic graphs with at most two exterior major vertices and terminal degree one.