\in X \mid \exists y\in{X}\ \text{such that}\ (x,y) \in
{\mathcal{Y}}_{1,\succeq{{\mathcal{R}}}}\ \text{or}\ (y,x) \in {\mathcal{Y}}_{1,\succeq_{{\mathcal{R}}}} \}.$$ If ${\mathcal{Y}}_{0,\succeq_{{\mathcal{R}}}}$ is the whole $X$, the social rule is said to be [*complete*]{}. A complete social rule is said to be [*strict*]{} if for each pair of social outcomes $x$ and $y$ the two conditions $x \succeq_{{\mathcal{R}}} y$ and $y \succeq_{{\mathcal{R}}} x$ are mutually exclusive (i.e. either the social outcome $x$ is preferred to the social outcome $y$ or the converse holds). For the sake of simplicity, we will consider only strict social rules. This restriction is almost always unnecessary, but it simplifies both the investigation and the presentation. Therefore, from now on, $\succ$ will always denote a complete strict social rule; unless explicitly stated, it will be considered as fixed. For the sake of shortness, we will always drop the words “complete” and “strict”.

#### The graph

The sets ${\mathcal{Y}}_{0,\succ}$ and ${\mathcal{Y}}_{1,\succ }$ are, respectively, the sets of nodes and arcs of a graph ${\mathcal{Y}}_{\succ}$. Two nodes $x$ and $y$ in ${\mathcal{Y}}_{0,\succ}$ are connected by an arc if $(x,y) \in {\mathcal{Y}}_{1,\succ}$ or $(y,x) \in {\mathcal{Y}}_{1,\succ}$; the orientation is from $x$ to $y$ in the former case and from $y$ to $x$ in the 