nodes is contained in a cycle. There is no bound on the length of the cycle, but every node of an irreducible tournament is contained in a $k$-cycle for all $k=3,4,\dotsc,M$. In particular, any irreducible tournament contains a hamiltonian cycle. A path is [*hamiltonian*]{} if it passes through all nodes. A tournament is [*transitive*]{} if it contains no cycle. Every tournament contains a hamiltonian path; if the tournament is transitive the hamiltonian path is unique.

An [*irreducible component*]{} ${{\cal T}}_i$ of ${{\cal T}}$ is a maximal irreducible sub-tournament of ${{\cal T}}$. The nodes of these irreducible components form a partition of the nodes of ${{\cal T}}$. Moreover, all the nodes of a component ${{\cal T}}_i$ either dominate or are dominated by all the nodes of another component ${{\cal T}}_j$. The transitive tournament ${\widetilde{{{\cal T}}}}$ whose nodes are the irreducible components of ${{\cal T}}$ and whose arcs are deduced by any arc between the two irreducible components (see Figure \[fig:derived\_tournament\]) is called the [*condensation*]{} of ${{\cal T}}$.

$\xymatrix@R=35pt{
        {\mbox{${{\cal T}}$}} & {*=<14pt>[o][F-]{p_5}} {\ar @{-} [ld] |-(0.57){\SelectTips{cm}{}\object@{>}}} {\ar @{-} [rd] |-(0.57){\SelectTips{cm}{}\object@{>}}} {\ar @{-} [d] |-(0.57){\SelectTips{cm}{}\object@{>}}}
          {\ar @{-} @/^55pt/ [dd] |-(.38){\SelectTips{cm}{}\object@{>}}}
          {\ar@{.} +UL(1.5);+UR(1.5) \ar@{.} +UR(1.5);+DR(1.5)
\ar@{.} +DR(1.5);+DL(1.5) \ar@{.} +DL(1.5);+UL(1.5) }{\ar@{.} +R(1.5);[rrr] |-{\object@{>}}} &
        