Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912666817134592 |
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| author | Bonomo-Braberman, Flavia Brettell, Nick Köhler, Noleen Munaro, Andrea Paulusma, Daniël |
| author_facet | Bonomo-Braberman, Flavia Brettell, Nick Köhler, Noleen Munaro, Andrea Paulusma, Daniël |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms Bonomo-Braberman, Flavia Brettell, Nick Köhler, Noleen Munaro, Andrea Paulusma, Daniël Computational Complexity Discrete Mathematics Combinatorics 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. |
| title | Non-crossing $H$-graphs: a generalization of proper interval graphs admitting FPT algorithms |
| topic | Computational Complexity Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2501.11192 |