graph $G$, a weight function ${\omega}$ and some number $t \in \mathbb N$, is $({\omega},t)$ an equidominating structure of $G$? This intractability result remains true in the seemingly simple case when $G$ is just the disjoint union of edges. That can be seen by applying literally the same reduction from the [Weak Partition]{} problem given by Milanič et al. [@MR2823204], who proved coNP-completeness for the the analogous question for maximal stable sets.

We remark that there is no characterization of the class of equidominating graphs in terms of forbidden induced subgraphs: if one attaches a pendant vertex to every vertex of an arbitrary graph (the so-called corona of a graph with $K_1$), the resulting graph is equidominating. The only existing result in this direction characterizes graphs that are both equidominating and domishold (see Theorem 2 in [@Payan1980]). Further, it is shown in [@Payan1980] that threshold graphs are equidominating.

To get a grip on the computational complexity of the problem, we introduce the following two parameterized notions of equidomination.

For a given $t\in{\ensuremath{\mathbb{N}}}$, a graph $G=(V,E)$ is called *target-$\boldsymbol{t}$ equidominating* if there is an equidominating structure of the form $({\omega},t)$ for $G$.

For a given $k\in{\ensuremath{\mathbb{N}}}$, a graph $G=(V,E)$ is called 