define the *quotient graph* $Q(G)$ of a graph $G$, where every twin class becomes a vertex and two vertices are adjacent if and only if the corresponding twin classes see each other. The twin partition and the quotient graph can be computed in linear time using one of the modular decomposition algorithms of [@ModDecomp1; @ModDecomp2; @ModDecomp3].

Properties of Twin Classes regarding Equidomination {#sec:propertiestwinclasses}
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In this section we use the twin relation to obtain structural results regarding equidomination. Initially, we introduce the following definition.

Let $G=(V,E)$ be a graph. Two vertices $x,y\in V$ are called *mds-exchangeable* if and only if there exists an mds $D\subseteq V$ with $|\{x,y\}\cap D|=1$ and if for all mds $D\subseteq V$ with $|\{x,y\}\cap D|=1$ the symmetric difference $(D \setminus\{x,y\}) \cup (\{x,y\} \setminus D)$ is also an mds.

Loosely speaking, two vertices are mds-exchangeable if they can be exchanged for another in any mds containing exactly one of them. Two mds-exchangeable vertices of an equidominating graph must have the same weight in every equidominating function. Even if two vertices are mds-exchangeable, they can both be elements of one mds (e.g. two non-adjacent vertices of $C_4$). While $c_1$, $c_2$ and $e$ in Figure \[fig:example\_equidom\_graph\] are mds-exchangeable, $s_1$ 