Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [30], form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption).As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those.Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of localitybased algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.
Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [29], form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption).As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those.Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem. * := a k+1 , we
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.
The generalised colouring numbers col r (G) and wcol r (G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications.In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number col r and a polynomial bound for the weak r-colouring number wcol r . In particular, we show that if G excludes K t as a minor, for some fixed t ≥ 4, then col r (G) ≤ t−1 2 (2r + 1) and wcol r (G) ≤ r+t−2 t−2 · (t − 3)(2r + 1) ∈ O(r t−1 ). In the case of graphs G of bounded genus g, we improve the bounds to col r (G) ≤ (2g + 3)(2r + 1) (and even col r (G) ≤ 5r + 1 if g = 0, i.e. if G is planar) and wcol r (G) ≤ 2g + r+2 2 (2r + 1).
We investigate the logical resources required to maintain knowledge about a property of a finite structure that undergoes an ongoing series of local changes such as insertion or deletion of tuples to basic relations. Our framework is closely related to the Dyn-FO-framework of Patnaik and Immerman and the FOIES-framework of Dong, Libkin, Su and Wong, and also builds on work of Weber and Schwentick. We assume that the dynamic process starts with an arbitrary, nonempty structure, but in contrast to previous work, we assume that, in general, structures are unordered. We show how to modify known dynamic algorithms for symmetric reachability, bipartiteness, k-edge connectivity and more, to work also without an order and with dynamic processes starting at an arbitrary graph. A history independent dynamic system (also called deterministic or memoryless) is one that maintains all auxiliary information independent of the update order. In 1997, Dong and Su posed the problem whether there exist history independent dynamic systems with FO-updates for symmetric reachability or bipartiteness. We give a positive answer to this question. We further show that there is a history independent system for tree isomorphism with FO+C-updates. On the other hand we show that on unordered structures first-order logic is too weak to maintain enough information to answer the equal cardinality query and the tree isomorphism query dynamically.
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