New Exact and Heuristic Algorithms for Graph Automorphism Group and Graph Isomorphism

Measures of graph symmetry, similarity, and identity have been extensively studied in graph automorphism and isomorphism detection problems. Nevertheless, graph isomorphism detection remains an open (unsolved) problem for many decades. In this paper, a new and efficient methodological paradigm, called optinalysis, is proposed for symmetry detections, similarity, and identity measures between isometric isomorphs or automorphs. Optinalysis is explained and expressed in clearly stated definitions and prove theorems, which conform to the definitions and theorems of isometry, isomorphism, and automorphism. Analogous to the polynomiality formalization for an efficient algorithm for graph isomorphism detection, optinalysis is however deterministic on polynomial and non-polynomial graph models.

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Graph Automorphism Group

IUPAC Standards Online ◽

10.1515/iupac.71.0356 ◽

2016 ◽

Author(s):

Vladimir I. Minkin

Keyword(s):

Automorphism Group ◽

Graph Automorphism

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Novel Techniques to Speed Up the Computation of the Automorphism Group of a Graph

Journal of Applied Mathematics ◽

10.1155/2014/934637 ◽

2014 ◽

Vol 2014 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

José Luis López-Presa ◽

Luis F. Chiroque ◽

Antonio Fernández Anta

Keyword(s):

Automorphism Group ◽

Classical Problem ◽

Search Tree ◽

Input Graph ◽

Graph Automorphism ◽

Ga Algorithm ◽

Speed Up ◽

Graph Families ◽

The Impact ◽

Asymptotic Time

Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. Most GA algorithms explore a search tree using the individualization-refinement procedure. Four novel techniques are proposed which increase the performance of any algorithm of this type by reducing the depth of the search tree and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of an input graph. Then, we describe how these techniques have been incorporated into the GA algorithm conauto, asconauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. Using a benchmark of different graph families, we have evaluated the impact of these techniques on the size of the search tree, observing a significant reduction both when they are applied individually and when all of them are applied together. This is also reflected in a reduction of the running time, which is substantial for some graph families. Finally, we have compared the search tree size of conauto-2.03 against those of other popular GA algorithms, observing that, in most cases, conauto explores less nodes than these algorithms.

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Symmetry of chemical structures: A novel method of graph automorphism group determination

Journal of Chemical Information and Modeling ◽

10.1021/ci00015a010 ◽

1993 ◽

Vol 33 (5) ◽

pp. 719-726 ◽

Cited By ~ 15

Author(s):

S. Bohanec ◽

M. Perdih

Keyword(s):

Automorphism Group ◽

Graph Automorphism ◽

Chemical Structures ◽

Novel Method

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A Moment-Based Method of Optinalysis

10.20944/preprints202008.0239.v1 ◽

2020 ◽

Author(s):

Kabir Bindawa Abdullahi

Keyword(s):

Unsolved Problem ◽

Graph Isomorphism ◽

Test Results ◽

Symmetry Detection ◽

Graph Automorphism ◽

Graph Models ◽

Moment Coefficient

Graph symmetry detection, similarity, and identity measures have been extensively studied in graph automorphism and isomorphism problems. Nevertheless, graph isomorphism and automorphism detection remain an open (unsolved) problem for many decades. In this paper, a new optinalytic coefficient termed as an optical moment coefficient was introduced for optinalysis. Its characteristic efficiency was tested for bijective property, invariance, deterministic polynomiality and non-polynomiality. The test results show that the new optical moment coefficient is very efficient for symmetry detections, similarity and identity measures between two isometric isomorphs and automorphs; and deterministic on polynomial and non-polynomial graph models.

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ON THE AUTOMORPHISM GROUP OF A FINITE P-GROUP WITH CYCLIC FRATTINI SUBGROUP

Mathematical Proceedings of the Royal Irish Academy ◽

10.3318/pria.2008.108.2.165 ◽

2008 ◽

Vol 108 (2) ◽

pp. 165-175 ◽

Cited By ~ 5

Author(s):

S. Fouladi ◽

A. R. Jamali ◽

R. Orfi

Keyword(s):

Automorphism Group ◽

Frattini Subgroup

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Comparison of Meta-heuristic Algorithms on Benchmark Functions

Academic Perspective Procedia ◽

10.33793/acperpro.02.03.41 ◽

2019 ◽

Vol 2 (3) ◽

pp. 508-517

Author(s):

FerdaNur Arıcı ◽

Ersin Kaya

Keyword(s):

Heuristic Algorithms ◽

Optimization Problems ◽

Search Algorithm ◽

Optimization Algorithms ◽

Gravitational Search Algorithm ◽

Differential Evolution Algorithm ◽

Population Based ◽

Time Interval ◽

Bee Colony ◽

Different Characteristics

Optimization is a process to search the most suitable solution for a problem within an acceptable time interval. The algorithms that solve the optimization problems are called as optimization algorithms. In the literature, there are many optimization algorithms with different characteristics. The optimization algorithms can exhibit different behaviors depending on the size, characteristics and complexity of the optimization problem. In this study, six well-known population based optimization algorithms (artificial algae algorithm - AAA, artificial bee colony algorithm - ABC, differential evolution algorithm - DE, genetic algorithm - GA, gravitational search algorithm - GSA and particle swarm optimization - PSO) were used. These six algorithms were performed on the CEC&rsquo;17 test functions. According to the experimental results, the algorithms were compared and performances of the algorithms were evaluated.

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