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41
Distortion invariant object recognition in the dynamic link architecture
 IEEE Transactions on Computers
, 1993
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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 190 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Isomorphism Testing for Embeddable Graphs Through Definability
 In Proceedings of 32nd ACM Symposium on Theory of Computing
, 1999
"... The kdimensional WeisfeilerLeman algorithm, for k 1, is a natural and simple combinatorial algorithm attempting to decide whether two given graphs are isomorphic. In this paper, we show that for every surface S (orientable or nonorientable) there is a k 1 such that the kdimensional WLalgorit ..."
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Cited by 26 (3 self)
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The kdimensional WeisfeilerLeman algorithm, for k 1, is a natural and simple combinatorial algorithm attempting to decide whether two given graphs are isomorphic. In this paper, we show that for every surface S (orientable or nonorientable) there is a k 1 such that the kdimensional WLalgorithm succeeds to decide isomorphism of graphs embeddable in S. To prove this, we use a close connection between the WLalgorithm and denability in certain nite variable logics that has been established by Cai, Furer, and Immerman [7]. 1. INTRODUCTION The graph isomorphism problem asks whether two given graphs are isomorphic. While complexity theoretic results indicate that the isomorphism problem is not NPcomplete (if it was, the polynomial hierarchy would collapse to its second level [6; 29]), no polynomial time algorithm for the general problem is known. However, there is a number of important classes of graphs on which the isomorphism problem is known to be solvable in polynomial t...
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 18 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Graph and map isomorphism and all polyhedral embeddings in linear time
 IN PROCEEDINGS OF THE 40TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. Th ..."
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Cited by 17 (5 self)
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k≥3, we want to find an embedding of G in S of face width at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 16 (6 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
FixedPoint Definability and Polynomial Time on Graphs with Excluded Minors
"... Abstract—We prove that fixedpoint logic with counting captures polynomial time on all classes of graphs with excluded minors. That is, for every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it i ..."
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Cited by 7 (2 self)
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Abstract—We prove that fixedpoint logic with counting captures polynomial time on all classes of graphs with excluded minors. That is, for every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it is definable in fixedpoint logic with counting. Furthermore, we prove that for every class C of graphs with excluded minors there is a k such that the kdimensional WeisfeilerLehman algorithm decides isomorphism of graphs in C in polynomial time. The WeisfeilerLehman algorithm is a combinatorial algorithm for testing isomorphism. It generalises the basic colour refinement algorithm and is much simpler than the known grouptheoretic algorithms for deciding isomorphism of graphs with excluded minors. The main technical theorem behind these two results is a “definable structure theorem ” for classes of graphs with excluded minors. It states that graphs with excluded minors can be decomposed into pieces arranged in a treelike structure, together with a linear order of each of the pieces. Furthermore, the decomposition and the linear orders on the pieces are definable in fixedpoint logic (without counting). Index Terms—descriptive complexity; graph minor theory; fixedpoint logic; graph canonisation I.
Motorcycle Graphs: Canonical Quad Mesh Partitioning
, 2008
"... We describe algorithms for canonically partitioning semiregular quadrilateral meshes into structured submeshes, using an adaptation of the geometric motorcycle graph of Eppstein and Erickson to quad meshes. Our partitions may be used to efficiently find isomorphisms between quad meshes. In addition ..."
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Cited by 7 (0 self)
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We describe algorithms for canonically partitioning semiregular quadrilateral meshes into structured submeshes, using an adaptation of the geometric motorcycle graph of Eppstein and Erickson to quad meshes. Our partitions may be used to efficiently find isomorphisms between quad meshes. In addition, they may be used as a highly compressed representation of the original mesh. These partitions can be constructed in sublinear time from a list of the extraordinary vertices in a mesh. We also study the problem of further reducing the number of submeshes in our partitions—we prove that optimizing this number is NPhard, but it can be efficiently approximated.
An Heuristic For Graph Symmetry Detection
 Proc. of Graph Drawing 99, Lecture Notes in Computer Science 1731:276285
, 1999
"... . We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introd ..."
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Cited by 6 (1 self)
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. We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introduction Testing whether a graph has any axial (rotational, central, respectively) symmetry is a NPcomplete problem [9]. Some restrictions (central symmetry with exactly one fixed vertex and no fixed edge) are polynomialy equivalent to the graph isomorphism test. Notice that this latter problem is not known to be either polynomial or NPcomplete in general. But several heuristics are known (e.g. [3]) and several restrictions leads to efficient algorithms: linear time isomorphism test for planar graphs [6] and interval graphs [8], polynomial time isomorphism test for fixed genus [10, 5], kcontractible graphs [12] and pairwise kseparable graphs [11], linear axial symmetry detection for plana...