FPT Isomorphism Test for $F$-Free Tournaments

We show that isomorphism of $F$-free tournaments can be solved in FPT time $f(k)\cdot n^{O(1)}$, where $k$ denotes the size of $F$, and $n$ denotes the size of the input tournaments. Our result extends on a previous FPT isomorphism test for tournaments of bounded twin-width [Grohe, Neuen 2024], as well as XP isomorphism tests parameterized by the VC dimension or the chromatic number [Raßmann, Schweitzer 2026]. It also implies that every non-trivial hereditary class of tournaments admits a polynomial-time isomorphism test. Our algorithm builds on a novel combination of spectral, geometric, combinatorial and group-theoretic tools.