Deciding First-Order Properties of Nowhere Dense Graphs
Deciding First-Order Properties of Nowhere Dense Graphs
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DOI：
10.1145/3051095
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发表时间：
2013-11
期刊：
影响因子：
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通讯作者：
Martin Grohe;S. Kreutzer;S. Siebertz
中科院分区：
文献类型：
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作者：
Martin Grohe;S. Kreutzer;S. Siebertz
Nešet松和Ossona de Mendez [2010，2011]引入的无处茂密的图形类别，形成了包括平面图等级的各种“稀疏图”，实际上是所有带有未成年人的类别的类别，以及有限的学位图和图形我们表明，在一阶逻辑中定义的图形的属性是固定参数在无处浓密的图表上（通过输入公式）至少对于在接受子图下关闭的图形类别，此结果是最佳的：对于所有在采用子图下关闭的图形C然后，C必须无处可（在合理的复杂性理论假设下）作为副产品扩展并改善了与未成年人的邻里覆盖物的覆盖范围，同时，我们的构造比我们的证明更简单。基于局部性的算法在逻辑方面，我们证明了Gaifman的局部定理的“排名”版本。
Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [2010, 2011], 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 (parameterized by the length of the input formula). 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 neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood 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 characterization 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.