& & {*=<14pt>[F-]{{{\cal T}}_3}} {\ar @{-} [d] |-(0.57){\SelectTips{cm}{}\object@{>}}} {\ar @{-} @/^15pt/ [dd] |-(.4){\SelectTips{cm}{}\object@{>}}}
        & {\mbox{${\widetilde{{{\cal T}}}}$}}
        \\
        {*=<14pt>[o][F-]{p_2}} {\ar @{-} [r] |-(0.57){\SelectTips{cm}{}\object@{>}}} {\ar @{-} [rd] |-(0.57){\SelectTips{cm}{}\object@{>}}} & {*=<14pt>[o][F-]{p_3}} {\ar @{-} [r] |-(0.57){\SelectTips{cm}{}\object@{>}}} {\ar @{-} [d] |-(0.57){\SelectTips{cm}{}\object@{>}}} & {*=<14pt>[o][F-]{p_4}}
          {\ar @{-} @/_15pt/ [ll] |-(.65){\SelectTips{cm}{}\object@{>}}} {\ar @{-} [ld] |-(0.57){\SelectTips{cm}{}\object@{>}}}
          {\ar@{.} [ll]+UL(1.5);[]+UR(1.5) \ar@{.} []+UR(1.5);[]+DR(1.5)
\ar@{.} []+DR(1.5);[ll]+DL(1.5) \ar@{.} [ll]+DL(1.5);[ll]+UL(1.5) } {\ar@{.} +R(1.5);[rr] |-{\object@{>}}}
        & & {*=<14pt>[F-]{{{\cal T}}_2}} {\ar @{-} [d] |-(0.57){\SelectTips{cm}{}\object@{>}}}
        \\
        & {*=<14pt>[o][F-]{p_1}} {\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@{>}}} &
        & & {*=<14pt>[F-]{{{\cal T}}_1}}
        }$

Without lack of generality, we will choose the subscripts so that if $i>j$ then ${{\cal T}}_i$ dominates ${{\cal T}}_j$. The maximal component of ${{\cal T}}$ will be denoted by ${{{\cal T}}_{\mathrm{MAX}}}$.

#### Score

The number of nodes dominated by a node $p$ is called the [*score*]{} of $p$. The sequence of the scores of the the nodes of a tournament is called the [*score sequence*]{} of ${{\cal T}}$. Up to a relabeling of the nodes, we can suppose that the score sequence of ${{\cal T}}$ is non-decreasing. A tournament is transitive if and only if its score sequence is $0,1,\dotsc,M-1$. A non-decreasing sequence $s_1,s_2,\dotsc,s_M$ of nonnegative integers is the score sequence of a 