Aalipour-Hafshejani et al.'s domination-polynomial uniqueness conjecture for complete bipartite graphs

Let $a,b\in\mathbb{N}$, and let ${\cal K}(a,b)$ denote the complete bipartite graph with parts of sizes $a$ and $b$. Two graphs are ${\cal D}$-equivalent when they have the same domination polynomial.

Aalipour-Hafshejani et al.'s conjecture. For all $a,b\in\mathbb{N}$, if $|a-b|\geq 2$, then ${\cal K}(a,b)$ is ${\cal D}$-unique; that is, every graph with the same domination polynomial as ${\cal K}(a,b)$ is isomorphic to it.

This conjecture concerns whether complete bipartite graphs with unequal parts differing by at least two are determined by their domination polynomial. The paper states that its results settle the conjecture affirmatively: any two such complete bipartite graphs with the same domination polynomial are isomorphic.