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Testing first-order properties for subclasses of sparse graphs
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Dvořák, Zdeněk, Králʼ, Daniel and Thomas, Robin (2013) Testing first-order properties for subclasses of sparse graphs. Journal of the ACM, Volume 60 (Number 5). Article number 36. doi:10.1145/2499483
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WRAP_Kral_1173667-ma-060613-databe.pdf - Accepted Version Download (635Kb) | Preview |
Official URL: http://dx.doi.org/10.1145/2499483
Abstract
We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed 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 deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO 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 simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Algorithms, Graph theory | ||||
| Journal or Publication Title: | Journal of the ACM | ||||
| Publisher: | Association for Computing Machinery, Inc. | ||||
| ISSN: | 0004-5411 | ||||
| Official Date: | October 2013 | ||||
| Dates: |
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| Volume: | Volume 60 | ||||
| Number: | Number 5 | ||||
| Page Range: | Article number 36 | ||||
| DOI: | 10.1145/2499483 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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