assertions hold:

1.  $|S_1|=|S_2|=2$, \[lem:SSCmitgleichemgewicht\_2elements\]

2.  ${\omega}$ is constant on $S_1\cup S_2$, \[lem:SSCmitgleichemgewicht\_const\]

3.  \[lem:SSCmitgleichemgewicht\_sameneighbors\] every twin class seen by $S_1$ is also seen by $S_2$ and vice versa.

Further, if two adjacent stable set classes of size two have the same closed neighborhood, all vertices of those stable set classes have the same weight in any equidominating structure.

Suppose one of the stable set classes has more than two elements, say $S_1$. We extend $S_1$ to an mds $D$ with $D\cap S_2=\emptyset$. Then, $D-x+y$ hat weight $t$ but following Observation \[obs:keinereineroderalleinSSC\] it is not an mds. So, assertion \[lem:SSCmitgleichemgewicht\_2elements\] is shown and let $S_1=\{x,x'\}$ and $S_2=\{y,y'\}$.

Let $D^*$ be an mds of $G[V\setminus N[S_1\cup S_2]]$. Then, $D\coloneqq D^*\cup\{x,y\}$ is an mds of G (note that $y'\in pn(x,D)$ and $x'\in pn(y,D)$). Observation \[obs:constantaufklassen\] yields assertion \[lem:SSCmitgleichemgewicht\_const\].

Suppose $S_2$ sees a vertex $v$ which is not seen by $S_1$. Extend $S_1\cup\{v\}$ to an mds $D^*$ such that $D^*\cap S_2=\emptyset$. Of course $D\coloneqq(D^*\setminus S_1) \cup S_2$ has weight $t$ but $D$ is not an mds, as $N[D]=N[D\setminus\{y\}]$. That proves assertion \[lem:SSCmitgleichemgewicht\_sameneighbors\].

The last statement of the lemma follows from the fact, that the vertices of the two adjacent stable set classes of 