$\boldsymbol{k}$*-equidominating* if there exists an equidominating structure $({\omega},t)$ with ${\omega}\colon V \rightarrow \{1,\ldots,k\}$ for some $t \in {\ensuremath{\mathbb{N}}}$. In this case, ${\omega}$ is said to be a $\boldsymbol{k}$*-equidominating function* and the pair $({\omega},t)$ a $\boldsymbol{k}$*-equidominating structure*.

Note that a $k$-equidominating graph is also $k'$-equidominating for all $k'\geq k$. It is clear that every target-$t$ equidominating graph is also $t$-equidominating since every vertex is contained in some mds and thus its weight, with respect to any equidominating function, cannot exceed $t$. The opposite, however, is not true. Indeed, the edgeless graph on $t+1$ vertices is $k$-equidominating for every $k\in {\ensuremath{\mathbb{N}}}$ but not target-$t$ equidominating.

In this paper, we study the following two parameterized versions of the [Equidomination]{} problem:

In the course of this paper we show that both the [$k$-Equidomination]{} problem and the [Target-$t$ Equidomination]{} problem are fixed-parameter tractable. We do this by using the so-called kernelization technique: we construct an equivalent instance the size of which is bounded by a function of the parameter. Further, we give FPT algorithms for both problems by applying an XP algorithm to the kernels. For a given graph on $n$ vertices and $m$ edges, this leads to a running time of $\mathcal{O}\left(nm^2 + n^2 