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Deciding firstorder properties for sparse graphs
"... We present a lineartime algorithm for deciding firstorder logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded treewidth, all proper minorclosed classes of graphs, graphs of bounded degree, graphs with no sub ..."
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Cited by 29 (1 self)
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We present a lineartime algorithm for deciding firstorder logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded treewidth, all proper minorclosed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost lineartime algorithm for deciding FOL properties in classes of graphs with locally bounded expansion; those include classes of graphs with locally bounded treewidth or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a lineartime initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low treedepth colorings.
Nonrepetitive Colouring via Entropy Compression
, 2012
"... A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively kchoosable if given lists of at least k colours at each vertex, there is a nonrepetitive colouringsuch that eachvertex iscolouredf ..."
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Cited by 12 (2 self)
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A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively kchoosable if given lists of at least k colours at each vertex, there is a nonrepetitive colouringsuch that eachvertex iscolouredfrom its own list. It is knownthat everygraph with maximum degree∆is c ∆ 2choosable, forsomeconstantc. We provethis result with c = 1 (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5choosable. The proofs of both these results are based on the MoserTardos entropycompression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that every graph with pathwidth k is nonrepetitively O(k 2)colourable.
Nonrepetitive Colourings of Planar Graphs with O(log n) Colours
, 2012
"... A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive kcolouring. Whether planar gr ..."
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Cited by 6 (3 self)
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A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive kcolouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O ( √ n) for nvertex planar graphs. We prove a O(log n) upper bound. 1
Layered Separators in MinorClosed Families with Applications
, 2013
"... Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω ( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minorclos ..."
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Cited by 3 (2 self)
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Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω ( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minorclosed families. We study a special type of graph separator, called a layered separator, which possibly has linear size in n, but has constant size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use these results to prove O(log n) bounds for a number of problems where O ( √ n) was a long standing previous best bound. This includes queuenumber and nonrepetitive chromatic number of bounded Euler genus graphs. We extend these results, with a log O(1) n bound, to all proper minorclosed families. This result also implies that every graph from a proper minorclosed class has a 3dimensional grid drawing in n log O(1) n volume, where the previous best bound was O(n 3/2). Only for planar graphs was a log O(1) n bound on the queuenumber previously known.
Structural Sparsity of Complex Networks: Bounded Expansion in Random Models and RealWorld Graphs
, 2014
"... This research aims to identify strong structural features of realworld complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded ex ..."
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This research aims to identify strong structural features of realworld complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded expansion—roughly, that any subgraph has bounded average degree after contracting disjoint boundeddiameter subgraphs—which formalizes the intuitive notion of “sparsity ” wellobserved in realworld complex networks. On the theoretical side, we analyze many previously proposed models for random networks and characterize, in very general scenarios, which produce graph classes of bounded expansion. We show that, with high probability, (1) Erdős–Rényi random graphs, generalized to have nonuniform edge probabilities and start from any boundeddegree graph, have bounded expansion; (2) the Molloy–Reed configuration model—matching any given degree sequence including “scalefree ” networks given by a powerlaw degree sequence—results in graphs of bounded expansion; and (3) the Kleinberg model and the Barabási–Albert model, in fairly typical setups, do not result in graphs of bounded expansion.
Hyperbolicity, degeneracy, and expansion of random intersection graphs
, 2014
"... We determine several key structural features of random intersection graphs, a natural model for many real world networks where connections are given by shared attributes. Notably, this model is mathematically tractable yet flexible enough to generate random graphs with tunable clustering. Specifical ..."
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We determine several key structural features of random intersection graphs, a natural model for many real world networks where connections are given by shared attributes. Notably, this model is mathematically tractable yet flexible enough to generate random graphs with tunable clustering. Specifically, we prove that in the homogeneous case, the model is logarithmically hyperbolic. Further, we fully characterize the degeneracy and the expansion of homogeneous random intersection graphs. Some of these results apply to simple inhomogeneous random intersection graphs. Finally, we comment on the algorithmic implications of these results.