[*dominate*]{} $q$. A [*sub-graph*]{} of $({{\cal V}},{{\cal E}})$ is a graph $({{\cal V}}',{{\cal E}}')$ such that ${{\cal V}}'\subset{{\cal V}}$ and ${{\cal E}}'\subset{{\cal E}}$.

A [*path*]{} ${P\left(p,q\right)}$ from $p$ to $q$ is a sequence of arcs of the type ${\overrightarrow{pp_1}},{\overrightarrow{p_1p_2}},\dotsc,{\overrightarrow{p_{k}q}}$. A [*domination path*]{} ${DP\left(p,q\right)}$ from $p$ to $q$ is a sequence of arcs of the type ${\overleftarrow{pp_1}},{\overleftarrow{p_1p_2}},\dotsc,{\overleftarrow{p_{k}q}}$. A [*cycle*]{} ${P\left(p,p\right)}$ (resp. a [*domination cycle*]{} ${DP\left(p,p\right)}$) is a path (resp. a domination path) from $p$ to itself. The [*length*]{} of a (domination) path is the number of arcs it contains; a cycle of length $k$ is called $k$-cycle.

#### Tournaments

A [*tournament*]{} is a complete graph (i.e. each pair of nodes $\{p,q\}$ is connected by an arc). By ${{\cal T}}$ we will always denote a tournament with $M$ nodes. A [*sub-tournament*]{} of ${{\cal T}}$ is a sub-graph of ${{\cal T}}$ that is a tournament.

A tournament is said to be [*reducible*]{} if it is possible to partition its nodes into two non-empty subsets ${{\cal V}}_1$ and ${{\cal V}}_2$ in such a way that all the nodes in ${{\cal V}}_1$ dominate all the nodes in ${{\cal V}}_2$; otherwise it is called [*irreducible*]{}. A tournament is irreducible if and only if each pair of 