We prove new parameterized complexity results for the FO Model Checking problem on a well-known generalization of interval and circular-arc graphs: the class of $H$-graphs, for any fixed multigraph $H$. In particular, we research how the parameterized complexity differs between two subclasses of $H$-graphs: proper $H$-graphs and non-crossing $H$-graphs, each generalizing proper interval graphs and proper circular-arc graphs. We first generalize a known result of Bonnet et al. (IPEC 2022) from interval graphs to $H$-graphs, for any (simple) forest $H$, by showing that for such $H$, the class of $H$-graphs is delineated. This implies that for every hereditary subclass ${\cal D}$ of $H$-graphs, FO Model Checking is in FPT if ${\cal D}$ has bounded twin-width and AW[$*$]-hard otherwise. As proper claw-graphs have unbounded twin-width, this means that FO Model Checking is AW[$*$]-hard for proper $H$-graphs for certain forests $H$ like the claw. In contrast, we show that even for every multigraph $H$, non-crossing $H$-graphs have bounded proper mixed-thinness and hence bounded twin-width, and thus FO Model Checking is in FPT on 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. It is known that a special case of FO Model Checking, Independent Set, is $\mathsf{W}[1]$-hard on $H$-graphs when parameterized by $\Vert H \Vert +k$, where $k$ is the size of a solution. We strengthen this $\mathsf{W}[1]$-hardness result to proper $H$-graphs. Hence, we solve, in two different ways, an open problem of Chaplick (Discrete Math. 2023), who asked about 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，在区间图和圆弧图的著名推广类——$H$图类（对任意固定多重图$H$）上证明了新的参数化复杂度结果。特别地，我们研究了$H$图的两个子类——真$H$图与非交叉$H$图（分别推广真区间图与真圆弧图）之间参数化复杂度的差异。首先，我们将Bonnet等人（IPEC 2022）关于区间图的结果推广至任意（简单）森林$H$对应的$H$图类，证明此类$H$图具有可界定性。这意味着对于$H$图的任意遗传子类${\cal D}$，当${\cal D}$具有有界孪生宽度时FO模型检测属于FPT，否则为AW[$*$]困难。由于真爪图具有无界孪生宽度，这表明对于某些森林$H$（如爪图），真$H$图上的FO模型检测是AW[$*$]困难的。与之相对，我们证明对任意多重图$H$，非交叉$H$图均具有有界真混合薄性，从而具有有界孪生宽度，因此当以$\Vert H \Vert+\ell$为参数时（其中$\Vert H \Vert$为$H$的规模，$\ell$为公式规模），非交叉$H$图上的FO模型检测属于FPT。已知FO模型检测的特例——独立集问题在$H$图上以$\Vert H \Vert +k$为参数时（$k$为解规模）是$\mathsf{W}[1]$困难的。我们将此$\mathsf{W}[1]$困难性结果强化至真$H$图类。由此，我们通过两种不同方式解决了Chaplick（Discrete Math. 2023）提出的公开问题：该问题探讨哪些计算问题在非交叉$H$图上比在真$H$图上具有更高效的求解算法。