A Simple and Efficient Algorithm for Determining Isomorphism of Planar Triply Connected Graphs

Weinberg, Louis

doi:10.1109/tct.1966.1082573

Cited by 86 publications

(54 citation statements)

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“…In an earlier paper [33], we developed a method to succinctly characterize the complete topology of a cell. That work was built on earlier work of Weinberg [58,59], who developed an efficient graph-theoretic algorithm to determine whether two triply-connected planar graphs are isomorphic. We showed that the edges and vertices of a cell can be treated as a planar graph, and Weinberg's method can then be used to calculate what we call a Weinberg vector for each cell.…”

Section: E Distribution Of Topological Typesmentioning

confidence: 99%

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks

et al. 2013

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Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250,000,000 cells to provide topological and geometrical statistics of this important class of networks. We also report correlations between some of these topological and geometrical measures. Using these results, we are able to corroborate several conjectures regarding the properties of three-dimensional Poisson-Voronoi networks and refute others. In many cases, we provide accurate fits to these data to aid further analysis. We also demonstrate that topological measures represent powerful tools for describing cellular networks and for distinguishing among different types of networks.

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Section: E Distribution Of Topological Typesmentioning

confidence: 99%

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks

et al. 2013

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“…In 1966, Weinberg [71] gave a very simple O(n 2 ) algorithm for the graph isomorphism problem of planar graphs. This was improved by Hopcroft and Tarjan [32,33] to O(n log n).…”

Section: The Graph Isomorphism Problemmentioning

confidence: 99%

“…Isomorphism of embedded graphs. We start by describing an easy O(n 2 ) algorithm based on the algorithm of Hopcroft and Tarjan [32], and Weinberg [71]. Then we expose a rather straightforward O(n log n) algorithm, which also modifies the algorithm by Hopcroft and Tarjan [33].…”

Section: Our Algorithmsmentioning

confidence: 99%

Graph and map isomorphism and all polyhedral embeddings in linear time

Kawarabayashi

Mohar

2008

Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing

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For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width 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 NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible 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 3-connected graphs.The second ingredient is a linear time algorithm for map isomorphism and Whitney equivalence. This part generalizes the seminal result of Hopcroft and Wong that graph isomorphism can be decided in linear time for planar graphs. The Graph Isomorphism ProblemThe graph isomorphism problem asks whether or not two given graphs are isomorphic. It is one of the most fundamental problems in the theory of algorithms and in complexity theory. It is probably the most notorious problem whose algorithmic complexity is still largely undecided. While some complexity theoretic results indicate that this problem is not NP-complete (if it were, the polynomial hierarchy would collapse to its second level, see [13,8,26,27,65]), no polynomial time algorithm is known for it, even with extended resources like randomization or quantum computing.On the other hand, there is a number of important classes of graphs on which the graph isomorphism problem is known to be solvable in polynomial time. For example, in 1990, Bodlaender [9] gave a polynomial time algorithm for graph isomorphism of graphs of bounded tree-width. Many NPhard problems can be solved in polynomial time, even linear time, when input is restricted to graphs of tree-width at most k [3,10]. So, Bodlaender's result may not be surprising, but the time complexity in [9] is O(n k+2 ), and no one could improve the time complexity to O(n O(1) ) so far. This indicates that even for graphs of bounded tree-width, the graph isomorphism problem is not trivial at all.In this paper, we are interested in planar graphs and, more generally, graphs of bounded genus. In 1966, Weinberg [71] gave a very simple O(n 2 ) algorithm for the graph isomorphism problem of planar graphs. This was improved by Hopcroft and ...

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“…To the best of our knowledge no such condition for general graphs exists that could replace the effort-consuming permutation of nodes following from the definition of isomorphism. Some works treating the isomorphism algorithms can be found in the reference list ( [2], [3], [4], [6], [11], [12])*.…”

Section: The Maximal Incidence Matrixmentioning

confidence: 99%

Search for a unique incidence matrix of a graph

Proskurowski

1974

BIT

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Abstract.This work deals with the problem of isomorphism of simple graphs without selfloops and gives a proposal for a canonical representation of graphs and an efficient deterministic procedure to find this representation.The article is disposed as follows : Section 1 defines the topological and computational language used and presents the isomorphism problem. The main results of the research are given in section 2, followed by the ALGOL-version of the algorithm and an analysis of its efficiency in section 3. The isomorphism problem.The problem, also known as the identification problem, is to decide whether two given graphs are isomorphic or not. Graph-theoretical notions and cornlrater representation.The topological language is based on the source reference of Harary [5]. Sometimes the presented approach has required minor alterations from the source's definition.A graph is defined as two finite sets of labels F (of nodes) and E (of edges) together with a function F (incidence function) that to some unordered pair of different nodes (vl,v~)e Y ascribes an edge e e E: e= F (vl, v~). The nodes v 1 and v~ will be called adjacent to each other and the edge e incident to both of them. Regarding the pair (vl, v~) as an unordered pair we refer to the notion of the unordered graph. When the function _F is simple-valued (with at most one edge corresponding to a pair of nodes) we refer to simple graphs.In the following we shall assume that the sets of nodes V and edges E consist of consecutive integers beginning with 1. One of the most common ways to represent a graph is by drawing an intuitive picture of it.Otherwise we shall represent a graph as a binary table of the incidence function, sometimes called the incidence matrix B. It has n = ]FI columns (corresponding to the nodes) and g= IEI rows, each corresponding to

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A Simple and Efficient Algorithm for Determining Isomorphism of Planar Triply Connected Graphs

Cited by 86 publications

References 5 publications

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks

Graph and map isomorphism and all polyhedral embeddings in linear time

Search for a unique incidence matrix of a graph

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