Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures -culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarský et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions.
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The complexity of the problem of deciding properties expressible in FO logic on graphs -the FO model checking problem (parameterized by the respective FO formula), is well-understood on so-called sparse graph classes, but much less understood on hereditary dense graph classes. Regarding the latter, a recent concept of twin-width [Bonnet et al., FOCS 2020] appears to be very useful. For instance, the question of these authors [CGTA 2019] about where is the exact limit of fixed-parameter tractability of FO model checking on permutation graphs has been answered by Bonnet et al. in 2020 quite easily, using the newly introduced twin-width. We prove that such exact characterization of hereditary subclasses with tractable FO model checking naturally extends from permutation to circle graphs (the intersection graphs of chords in a circle). Namely, we prove that under usual complexity assumptions, FO model checking of a hereditary class of circle graphs is in FPT if and only if the class excludes some permutation graph. We also prove a similar excluded-subgraphs characterization for hereditary classes of interval graphs with FO model checking in FPT, which concludes the line a research of interval classes with tractable FO model checking started in [Ganian et al., ICALP 2013]. The mathematical side of the presented characterizations -about when subclasses of the classes of circle and permutation graphs have bounded twin-width, moreover extends to so-called bounded perturbations of these classes. ACM Subject ClassificationMathematics of computing → Graph theory; Theory of computation → Fixed parameter tractability Keywords and phrases twin-width, FO model checking, circle graph, interval graph, FPT Funding Supported by the Czech Science Foundation, project no. 20-04567S. The Dagstuhl seminar 21391 Sparsity in Algorithms, Combinatorics and Logic (September 2021) significantly contributed to part of these results.
While structural width parameters (of the input) belong to the standard toolbox of graph algorithms, it is not the usual case in computational geometry. As a case study we propose a natural extension of the structural graph parameter of clique-width to geometric point configurations represented by their order type. We study basic properties of this clique-width notion, and relate it to the monadic second-order logic of point configurations. As an application, we provide several linear FPT time algorithms for geometric point problems which are NP-hard in general, in the special case that the input point set is of bounded clique-width and the clique-width expression is also given.
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