
Fiding forbidden minors in sublinear time: a O(n^1/2 + o(1))query onesided tester for minor closed properties on bounded degree graphs
Let G be an undirected, bounded degree graph with n vertices. Fix a fini...
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Property Testing of Planarity in the CONGEST model
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Testing Hamiltonicity (and other problems) in MinorFree Graphs
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Testable Properties in General Graphs and Random Order Streaming
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Faster sublinear approximations of kcliques for low arboricity graphs
Given query access to an undirected graph G, we consider the problem of ...
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Adapting the Directed Grid Theorem into an FPT Algorithm
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the mo...
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Inapproximability of Additive Weak Contraction under SSEH and Strong UGC
Succinct representations of a graph have been objects of central study i...
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Finding forbidden minors in sublinear time: a n^1/2+o(1)query onesided tester for minor closed properties on bounded degree graphs
Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove ε n edges from G to make it Hminor free (for some small constant ε > 0). We give an n^1/2+o(1)time randomized procedure that, with high probability, finds an Hminor in such a graph. As an application, suppose one must remove ε n edges from a bounded degree graph G to make it planar. This result implies an algorithm, with the same running time, that produces a K_3,3 or K_5 minor in G. No prior sublinear time bound was known for this problem. By the graph minor theorem, we get an analogous result for any minorclosed property. Up to n^o(1) factors, this resolves a conjecture of BenjaminiSchrammShapira (STOC 2008) on the existence of onesided property testers for minorclosed properties. Furthermore, our algorithm is nearly optimal, by an Ω(√(n)) lower bound of Czumaj et al (RSA 2014). Prior to this work, the only graphs H for which nontrivial onesided property testers were known for Hminor freeness are the following: H being a forest or a cycle (Czumaj et al, RSA 2014), K_2,k, (k× 2)grid, and the kcircus (Fichtenberger et al, Arxiv 2017).
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