Irreducible Pairings and Indecomposable Tournaments

We only consider finite structures. With every totally ordered set V and a subset P of \(\left( {\begin{array}{c}V\\ 2\end{array}}\right) \), we associate the underlying tournament \(\textrm{Inv}({\underline{V}}, P)\) obtained from the transitive tournament \({\underline{V}}:=(V, \{(x,y) \in V \times V: x < y \})\) by reversing P, i.e., by reversing the arcs (x, y) such that \(\{x,y\} \in P\). The subset P is a pairing (of \(\cup P\)) if \(|\cup P| = 2|P|\), a quasi-pairing (of \(\cup P\)) if \(|\cup P| = 2|P|-1\); it is irreducible if no nontrivial interval of \(\cup P\) is a union of connected components of the graph \((\cup P, P)\). In this paper, we consider pairings and quasi-pairings in relation to tournaments. We establish close relationships between irreducibility of pairings (or quasi-pairings) and indecomposability of their underlying tournaments under modular decomposition. For example, given a pairing P of a totally ordered set V of size at least 6, the pairing P is irreducible if and only if the tournament \(\textrm{Inv}({\underline{V}}, P)\) is indecomposable. This is a consequence of a more general result characterizing indecomposable tournaments obtained from transitive tournaments by reversing pairings. We obtain analogous results in the case of quasi-pairings.