We prove that isomorphism of tournaments of twin width at most $k$ can be decided in time $k^{O(\log k)}n^{O(1)}$. This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters. Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is $o(\sqrt n)$. (Throughout this abstract, $n$ is the order of the input graphs.)

翻译：我们证明，对于孪生宽度至多为$k$的锦标赛，其同构性可在时间$k^{O(\log k)}n^{O(1)}$内判定。这意味着，对于有界或适度增长的孪生宽度的锦标赛类，同构问题是多项式时间可解的。相比之下，存在有界孪生宽度的无向图类，其同构问题是完全的，即这些图类的同构问题与一般图同构问题难度相当。孪生宽度是一种图参数，近期才被引入（Bonnet等人，FOCS 2020），但此后在结构图论中受到广泛关注。在有向图上，它在功能上比团宽度更小。我们证明，在锦标赛（而非一般有向图）上，它在功能上也比有向树宽度更小（因此，对割宽度和有向路径宽度同样成立）。因此，我们的结果表明，当以这些参数作为参数时，锦标赛同构测试也是固定参数可处理的。我们的同构算法大量采用了群论技术。这似乎是必要的：作为第二个主要结果，我们证明组合性韦斯法伊勒-莱曼算法无法判定孪生宽度至多为35且维度为$o(\sqrt n)$的锦标赛的同构性。（在整个摘要中，$n$是输入图的阶数。）