Tournaments that omit N 5 are well - quasi - ordered by Brenda

The tournament N5 can be obtained from the transitive tournament on {1, . . . , 5}, with the natural order, by reversing the edges between successive vertices. Tournaments that do not have N5 as a subtournament are said to omit N5. We describe the structure of tournaments that omit N5 and use this with Kruskal’s Tree Theorem to prove that this class of tournaments is well-quasi-ordered. The proof involves an encoding of the indecomposable tournaments omitting N5 by a finite alphabet. The main technical problem is giving an explicit description of the indecomposable tournaments omitting N5. 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 N5 will also be found in our list.