two flags for each vertex and a global counter we can do this considering every edge two times. So, line 3 needs $\mathcal O(m)$ time which leads to a total running time of $\mathcal O(nm)$.

To discover all bundles one has to apply Algorithm \[alg:adjacentmdsexchangeable\] for every edge, which gives us a total running time of $\mathcal O(nm^2)$ to determine the pseudo class partition.

An XP algorithm for the *k*-[Equidomination]{} Problem {#sec:xpalgorithm}
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In this section we describe an algorithm which decides whether a given graph is $k$-equidominating for some fixed $k\in {\ensuremath{\mathbb{N}}}$ with a running with only $k$ appearing in the exponents. The aim is to apply this algorithm to the constructed kernels of the parameterized problems. The algorithm mainly follows the ideas and the algorithm for the $k$-[Equistability]{} problem of Levit et al. [@kequistable_fpt; @Recog_kequistable]. However, it has to be extended due to the existence of clique bundles and stable set bundles.

The basic idea of the algorithm is that by considering the pseudo classes one does not have to examine every possible weight function nor every possible subset of vertices. Since different vertices of the same pseudo class, roughly said, play the same role regarding domination, two 