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

TL;DR

The paper studies first-order (FO) model checking on intersection-graph generalizations called $H$-graphs, focusing on two hereditary subfamilies: proper $H$-graphs and non-crossing $H$-graphs. It shows that when $H$ is a forest, the $H$-graph class is delineated, leading to an AW[*]-hardness vs FPT dichotomy for hereditary subclasses via bounded twin-width, and that non-crossing $H$-graphs have bounded proper mixed-thinness for any multigraph $H$, hence bounded twin-width, which yields FPT FO Model Checking parameterized by $\Vert H \Vert+\ell$ given a non-crossing $H$-representation; meanwhile, Independent Set is $\mathsf{W}[1]$-hard on proper $H$-graphs, strengthening prior results. Altogether, the work resolves Chaplick's question about problems being faster on non-crossing $H$-graphs than on proper $H$-graphs, and highlights non-crossing $H$-graphs as a robust generalization with favorable algorithmic properties.

Abstract

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.

Figures (2)

• Figure 1: Four different representations of the claw ($K_{1,3}$) as a claw-graph.
• Figure 2: A $3$-thin representation of a graph. The vertices are ordered increasingly by their $y$-coordinate, and each class corresponds to a column of vertices.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Theorem 5
• Corollary 5
• Corollary 6
• Theorem 7
• Corollary 8
• Theorem 9