Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms

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.