class is exactly one of following: a) a singleton class, b) a stable set class, c) a clique class, d) a stable set bundle or e) a clique bundle. By this definition and the previous discussion we get the following result.

\[cor:unique\_pseudo\_class\_partition\] There is a unique partition of the vertices of a graph into pseudo classes.

The introduction of pseudo classes is motivated by the following corollary, which is a summary of this section.

\[cor:differentpseudodifferentweights\] Let $G$ be an equidominating graph with equidominating structure $({\omega},t)$ and $P_1,P_2$ be two different pseudo classes. Then for all $x\in P_1$ and $y\in P_2$ it holds that ${\omega}(x)\neq{\omega}(y)$.

To recognize bundles we developed Algorithm \[alg:adjacentmdsexchangeable\], which decides for two adjacent vertices whether or not they are mds-exchangeable. More precisely, the algorithm checks if there can be private neighbors of one vertex in any dominating set that are not seen by the other one. After computing the twin partition one can apply Algorithm \[alg:adjacentmdsexchangeable\] to adjacent clique classes and singleton classes, and to adjacent stable set classes of size two to find all clique bundles and stable set bundles, respectively.

return **NO** return **YES**

Algorithm \[alg:adjacentmdsexchangeable\] is correct and runs in $\mathcal O(nm)$ time.

First 