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., J. ACM 2022), 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 $o(n)$. (Throughout this abstract, $n$ is the order of the input graphs.)

翻译：我们证明了双宽度至多为$k$的竞赛图的同构问题可以在$k^{O(\log k)}n^{O(1)}$时间内判定。这意味着对于双宽度有界或适度增长的竞赛图类，其同构问题属于多项式时间可解。相比之下，存在双宽度有界的无向图类是同构完全的，即这些图类的同构问题与一般图同构问题难度相当。双宽度是近年来新引入的图参数（Bonnet等人，J. ACM 2022），此后在结构图论中受到广泛关注。在有向图上，该参数在函数意义上小于团宽度。我们证明在竞赛图（但非一般有向图）上，该参数在函数意义上也小于有向树宽度（因此同样适用于割宽度和有向路径宽度）。由此可见，当以这些参数中的任意一个作为参数时，竞赛图同构测试也属于固定参数可解问题。我们的同构算法大量运用了群论技术，这似乎是必要的：作为第二个主要结果，我们证明当维度为$o(n)$时，组合式Weisfeiler-Leman算法无法判定双宽度至多为35的竞赛图的同构性。（本摘要中$n$均指输入图的阶数。）