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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 2, Pages 189–203 (Mi rcd74)

Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms

Vyacheslav Z. Grinesa, Dmitry S. Malyshevab, Olga V. Pochinkaa, Svetlana Kh. Zininac

a National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155, Russia
b N. I. Lobachevsky State University of Nizhni Novgorod, ul. Gagarina 23, Nizhny Novgorod, 603950, Russia
c Ogarev Mordovia State University, ul. Bolshevistskaya 68, Saransk, 430005, Russia

Abstract: It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.

Keywords: Morse – Smale diffeomorphism, gradient-like diffeomorphism, topological classification, three-color graph, directed graph, graph isomorphism, surface orientability, surface genus, polynomial-time algorithm, magnetic field

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01- 03687-a16-31-60008-mol_a_dk Ministry of Education and Science of the Russian Federation MK-4819.2016.1 Russian Science Foundation 14-11-00044 This work was supported by the Russian Foundation for Basic Research (grants 15-01-03687-a, 16-31-60008-mol_a_dk), Russian Science Foundation (grant 14-11-00044), the Basic Research Program at the HSE (project 98) in 2016, by LATNA laboratory, National Research University Higher School of Economics, and by RF President grant MK-4819.2016.

DOI: https://doi.org/10.1134/S1560354716020040

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Bibliographic databases:

MSC: 32S50, 37C15
Accepted:04.02.2016
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Citation: Vyacheslav Z. Grines, Dmitry S. Malyshev, Olga V. Pochinka, Svetlana Kh. Zinina, “Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms”, Regul. Chaotic Dyn., 21:2 (2016), 189–203

Citation in format AMSBIB
\Bibitem{GriMalPoc16} \by Vyacheslav Z. Grines, Dmitry S. Malyshev, Olga V. Pochinka, Svetlana Kh. Zinina \paper Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 2 \pages 189--203 \mathnet{http://mi.mathnet.ru/rcd74} \crossref{https://doi.org/10.1134/S1560354716020040} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3486005} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000374286800004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84963748109} 

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• http://mi.mathnet.ru/eng/rcd/v21/i2/p189

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Sistemy Morsa–Smeila i topologicheskaya struktura nesuschikh mnogoobrazii”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 5–40
2. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds”, Russian Math. Surveys, 74:1 (2019), 37–110