Abstract
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 on 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 on dense classes of graphs or other structures. We study the FO model checking problem on 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 on posets of bounded width. We also complement the tractability results by related hardness reductions.
Highlights
Algorithmic meta-theorems are results stating that all problems expressible in a certain language are efficiently solvable on certain classes of structures, e.g. of finite graphs
Inspired by the geometric case of interval graphs, we propose to study dense graph classes defined in geometric terms, such as intersection and visibility graphs, with respect to tractability of their FO model checking problem
We have identified several FP tractable cases of the FO model checking problem of geometric graphs, and complemented these by hardness results showing quite strict limits of FP tractability on the studied classes
Summary
Algorithmic meta-theorems are results stating that all problems expressible in a certain language are efficiently solvable on certain classes of structures, e.g. of finite graphs. Inspired by the geometric case of interval graphs, we propose to study dense graph classes defined in geometric terms, such as intersection and visibility graphs, with respect to tractability of their FO model checking problem. Intersection and visibility graphs present natural examples of non-monotone somewhere dense graph classes to which the great “sparse” FO tractability result of [20] cannot be (at least not ) applied Their supplementary geometric structure allows to better understand (as we have seen already in [16]) the boundaries of tractability of FO model checking on them, which is, to current knowledge, terra incognita for hereditary graph classes in general. We emphasize the seemingly simple tractable case (Corollary 4.2) of permutation graphs of bounded clique size: in relation to so-called stability notion (cf [1]), already the hereditary class of triangle-free permutation graphs has the n-order property (i.e., is not stable), and yet FO model checking of this class is FPT. The statements with removed proofs are marked by * and they can be found, for example, in the arXiv version
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