Abstract
In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas’s invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time.
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Notes
The notion of an effectively solvable problem rises to Cobham (1964), who asserts that a computational problem can be feasibly computed on some device only if it can be computed in time bounded by a polynomial on a parameter representing the length of input data. The complexity status of the general graph isomorphism problem is not proved to be polynomial or not. Graphs in the criteria are not graphs of the general type, they possess peculiar combinatorial properties.
If the non-wandering set includes also a finite number of hyperbolic periodic orbits, then the flow is called a Morse-Smale flow and topological classification of such flows on surfaces has been obtained, for example, in Peixoto (1973) and Oshemkov and Sharko (1998). If a flow allows also trajectories connecting saddle points, then such flow is called an \(\Omega \)-stable flow; classification of \(\Omega \)-stable flows without limit cycles on surfaces has been obtained in Kruglov et al. (2018a); classification of all \(\Omega \)-stable flows on surfaces has been obtained in Kruglov et al. (2018b).
All gradient-like flows without saddle points are given on a sphere and have only two fixed points: source and sink. All such flows are topologically conjugate.
The classification results in Grines et al. (2016) have been obtained for gradient-like diffeomorphisms, but they are also valid for flows.
It can be directly checked that the property of the permutation to be a power of a cyclic permutation is independent on a choice of the curves \(c_\omega \) and \(c_{\omega '}\).
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Acknowledgements
The algorithm for the Peixoto invariants was created with a support of the Russian Foundation for Basic Research Project 18-31-00022 mol\(\underline{\,\,\,}\)a. The polynomial-time algorithm for the Wang invariant was obtained as an output of the research project of the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019. The algorithms for the Fleitas’ graph invariant was created under a support of RF President Grant MD-879.2019.1 and Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics. The polynomial-time algorithm for the Oshemkov-Sharko invariant and conjugacy theorem were implemented in the framework of the Russian Science Foundation Project 17-11-01041.
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Kruglov, V.E., Malyshev, D.S. & Pochinka, O.V. On Algorithms that Effectively Distinguish Gradient-Like Dynamics on Surfaces. Arnold Math J. 4, 483–504 (2018). https://doi.org/10.1007/s40598-019-00103-0
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DOI: https://doi.org/10.1007/s40598-019-00103-0