First-Order Logic and Twin-Width for Some Geometric Graphs

TL;DR

The paper investigates when first-order model checking remains tractable on geometric graph classes by linking tractability to the twin-width parameter via delineation. It establishes delineation for circular-arc graphs and axis-parallel unit-segment graphs (APUS) by reducing to matrix grids and transversal-pair obstructions, and it shows non-delineation for 1.5D-terrain visibility graphs. The authors develop a framework using ordered representations, transductions, and a regional square-splitting technique to transfer bounded twin-width from local to global structures. They also construct explicit obstructions demonstrating that delineation can fail under relaxed conditions, highlighting that bounded merge-width can still yield FPT FO model checking even when twin-width is unbounded. Overall, the work clarifies the boundary between tractable and intractable FO logic on geometric intersection graphs and connects geometric representations to algorithmic consequences via twin-width and merge-width tools.

Abstract

For some geometric graph classes, tractability of testing first-order formulas is precisely characterised by the graph parameter twin-width. This was first proved for interval graphs among others in [BCKKLT, IPEC '22], where the equivalence is called delineation, and more generally holds for circle graphs, rooted directed path graphs, and $H$-graphs when $H$ is a forest. Delineation is based on the key idea that geometric graphs often admit natural vertex orderings, allowing to use the very rich theory of twin-width for ordered graphs. Answering two questions raised in their work, we prove that delineation holds for intersection graphs of non-degenerate axis-parallel unit segment graphs, but fails for visibility graphs of 1.5D terrains. We also prove delineation for intersection graphs of circular arcs.

Figures (6)

• Figure 1: A transversal pair $T_3$.
• Figure 3: An instance being split along unit squares (drawn in gray). The cyan segments are the set $\mathcal{H}_{s-1,t}$, while the magenta segments are the set $\mathcal{V}_{s,t}$. Together, they form $\mathcal{F}_{s,t,{\scalerel*{}{1}}}$.
• Figure 5: The graph $H_\sigma^5$ for the permutation $\sigma = 147258369$.
• Figure 7: A degenerate APUS whose intersection graph is $H_\sigma^5$ for $\sigma = 147258369$.
• Figure 9: Construction of non-delineated terrain visibility graphs. Dotted lines represent the 'horizon' for each $b_i$, which can be controlled by moving $b_i$ up or down. This allows the half-graph between the $b_i$s and $c_j$s to represent any permutation of the $b_i$s, here $369258147$.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Conjecture 5: gajarsky2020FO
• Theorem 6: Ramsey theorem for grids
• Lemma 7
• Lemma 8
• Theorem 9: Bonnet2022twinwidth1twin-width4
• Theorem 10: twin-width4