+ t^{2t^2+3t+1}\right)$ for the [Target-$t$ Equidomination]{} problem and $\mathcal{O}\left(nm^2 + n^2 + k^{6k^2+7k+1}\right)$ for the [$k$-Equidomination]{} problem.

This paper is structured as follows: in Section \[sec:preliminaries\] we give a brief explanation of the needed fundamentals. In Section \[sec:propertiestwinclasses\] we examine relationships between the twin relation and equidomination and give an algorithm to discover certain structures that can appear. We develop an XP algorithm for the [$k$-Equidomination]{} problem (which can also be used for the [Target-$t$ Equidomination]{} problem) in Section \[sec:xpalgorithm\]. We make use of this algorithm and of reduction rules described in Section \[sec:reduction\] to deduce the desired tractability results, stated in Section \[sec:kernelkequidomination\]. Afterward, we characterize the class of hereditarily equidominating graphs in Section \[sec:hereditarily\]. In Section \[sec:conclusion\] we draw a conclusion and give a brief outlook.

Preliminaries {#sec:preliminaries}
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In the first part of this section we give some basic definitions and notations that we use in this paper. In the second part we give a brief introduction to fixed-parameter tractability and the third part is devoted to the so-called twin relation.

Basic Definitions and Notations
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The graphs we work with all are undirected, simple and finite. For a vertex $v\in V$ of a graph $G=(V,E)$ we 