A separator theorem for graphs with an excluded minor and its applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
Linear time algorithm for isomorphism of planar graphs (Preliminary Report)
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Randomized minimum spanning tree algorithms using exponentially fewer random bits
ACM Transactions on Algorithms (TALG)
Every minor-closed property of sparse graphs is testable
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Fast Distributed Approximations in Planar Graphs
DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs
SIAM Journal on Computing
Local Graph Partitions for Approximation and Testing
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Parameter testing in bounded degree graphs of subexponential growth
Random Structures & Algorithms
Testing outerplanarity of bounded degree graphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Every property of hyperfinite graphs is testable
Proceedings of the forty-third annual ACM symposium on Theory of computing
An efficient partitioning oracle for bounded-treewidth graphs
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
New sublinear methods in the struggle against classical problems
New sublinear methods in the struggle against classical problems
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Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient partition oracles. A partition oracle is a procedure that, given access to the incidence lists representation of a bounded-degree graph G=(V,E) and a parameter ε, when queried on a vertex v∈V, returns the part (subset of vertices) which v belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most ε|V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/ε, thus improving on the result of Hassidim et al. (Proceedings of FOCS 2009) who gave a partition oracle with query complexity exponential in 1/ε. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.