We prove new parameterized complexity results for the FO Model Checking problem and in particular for Independent Set, for two recently introduced subclasses of $H$-graphs, namely proper $H$-graphs and non-crossing $H$-graphs. It is known that proper $H$-graphs, and thus $H$-graphs, may have unbounded twin-width. However, we prove that for every connected multigraph $H$ with no self-loops, non-crossing $H$-graphs have bounded proper mixed-thinness, and thus bounded twin-width. Consequently, we can apply a well-known result of Bonnet, Kim, Thomass\'e, and Watrigant (2021) to find that the FO Model Checking problem is in $\mathsf{FPT}$ for non-crossing $H$-graphs when parameterized by $\Vert H \Vert+\ell$, where $\Vert H \Vert$ is the size of $H$ and $\ell$ is the size of a formula. In particular, this implies that Independent Set is in $\mathsf{FPT}$ on non-crossing $H$-graphs when parameterized by $\Vert H \Vert+k$, where $k$ is the solution size. In contrast, Independent Set for general $H$-graphs is $\mathsf{W[1]}$-hard when parameterized by $\Vert H \Vert +k$. We strengthen the latter result by proving thatIndependent Set is $\mathsf{W[1]}$-hard even on proper $H$-graphs when parameterized by $\Vert H \Vert+k$. In this way, we solve, subject to $\mathsf{W[1]}\neq \mathsf{FPT}$, an open problem of Chaplick (2023), who asked whether there exist problems that can be solved faster for non-crossing $H$-graphs than for proper $H$-graphs.

翻译：我们针对FO模型检测问题，特别是独立集问题，在最近引入的两类$H$-图子类——真$H$-图与非交叉$H$-图上证明了新的参数化复杂度结果。已知真$H$-图（因而一般$H$-图）可能具有无界孪生宽度。然而，我们证明对于任意无自环的连通多重图$H$，非交叉$H$-图具有有界真混合薄性，从而具有有界孪生宽度。因此，我们可以应用Bonnet、Kim、Thomassé与Watrigant（2021）的著名结果，得出FO模型检测问题在非交叉$H$-图上属于$\mathsf{FPT}$（参数化为$\Vert H \Vert+\ell$，其中$\Vert H \Vert$为$H$的规模，$\ell$为公式规模）。特别地，这意味着独立集问题在非交叉$H$-图上参数化为$\Vert H \Vert+k$（$k$为解规模）时属于$\mathsf{FPT}$。相比之下，一般$H$-图上的独立集问题在参数化为$\Vert H \Vert+k$时是$\mathsf{W[1]}$-难的。我们通过证明独立集问题即使在真$H$-图上参数化为$\Vert H \Vert+k$时也是$\mathsf{W[1]}$-难的，从而强化了后一结果。由此，我们在$\mathsf{W[1]}\neq \mathsf{FPT}$的假设下，解决了Chaplick（2023）提出的一个开放问题：是否存在某些问题在非交叉$H$-图上比在真$H$-图上可更快求解。