The main technical problem is giving an explicit
description of the indecomposable tournaments omitting $N_5$. The key to
the proof that the explicit description is complete is the observation
that for any indecomposable tournament $T$ with $n>1$ vertices, there is a
proper indecomposable subtournament of $T$ with $n-2$ or $n-1$ vertices.
Thus the claim can be verified by a natural inductive procedure; it
suffices to check that for any tournament $T$ in the explicitly given
list, any indecomposable extension of $T$ by at most 2 vertices that omits
$N_5$ will also be found in our list.
Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2002/2002-31.ps.gz