Graphs and flows on surfaces

Nikolaev, Igor

doi:10.1017/s0143385798097636

Cited by 13 publications

(6 citation statements)

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“…Two cyclic graphs are isomorphic as cyclic graphs if there is a graph isomorphism between the underlying graphs which respects the cyclic orders. These objects are called`graphs with rotation systems' in [18,19]; in [18] it is shown that a certain class of directed cyclic graphs classi®es Morse±Smale¯ows up to topological equivalence. A similar result for foliations is proved in [19].…”

Section: Cyclic Graphsmentioning

confidence: 99%

“…Suppose that R is a commutative ring, and that x P R. For each i, j, as the map C i j takes values in ZX , we can compose it with the natural ring homomorphism from ZX to R mapping X to x, obtaining a map C i j x from G Ã to R. Note that in®nite sums of the functions C i j x make sense, as only ®nitely many are nonzero on any given cyclic graph. Furthermore, when f does satisfy (17), the l i j are uniquely determined by (18).…”

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confidence: 99%

“…Theorem 1 implies that each C i j x satis®es (17). Thus if f has the form (18), then f satis®es (17).…”

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confidence: 99%

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A Polynomial Invariant of Graphs On Orientable Surfaces

Bollobás

Riordan

2001

Proceedings of the London Mathematical Society

192

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We consider cyclic graphs, that is, graphs with cyclic orders at the vertices, corresponding to 2‐cell embeddings of graphs into orientable surfaces, or combinatorial maps. We construct a three variable polynomial invariant of these objects, the cyclic graph polynomial, which has many of the useful properties of the Tutte polynomial. Although the cyclic graph polynomial generalizes the Tutte polynomial, its definition is very different, and it depends on the embedding in an essential way. 2000 Mathematical Subject Classification: 05C10.

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Section: Cyclic Graphsmentioning

confidence: 99%

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confidence: 99%

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A Polynomial Invariant of Graphs On Orientable Surfaces

Bollobás

Riordan

2001

Proceedings of the London Mathematical Society

192

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“…The possible shapes for canonical regions of Morse-Smale fields are shown on Fig. 9, see also [N,Sec. 1.2].…”

Section: Flexibility Of Automorphismsmentioning

confidence: 99%

Global Bifurcations in Generic One-Parameter Families with a Parabolic Cycle on S 2

Goncharuk¹,

Ilyashenko²,

Solodovnikov³

2019

MMJ

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We classify global bifurcations in generic one-parameter local families of vector fields on S 2 with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.In this article, we consider one-parameter families of vector fields on B × S 2 . Here B ⊂ R is the base of the family. We work with local families with bases (R, 0), in the following sense. Definition 3.A local family of vector fields at ε = 0 is a germ at {0} × S 2 of a family onDefinition 4. An unfolding of a vector field v is a local family for which v corresponds to zero parameter value. We say that this family unfolds the vector field.The following definition lists the notions of equivalence for local families of vector fields that we will use in this paper. Definition 5. Let B, B be topological balls in R that contain 0. Two local families of vector fields {v α , α ∈ B}, {w β , β ∈ B } on S 2 are called

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“…It's known that Lyapunov graphs are not complete invariant for Morse-Smale flows (i.e. there are Morse-Smale flows with isomorphic Lyapunov graphs but which are not topologically equivalent) but that Peixoto graphs are complete invariant for Morse-Smale flows [9]. In [11], non-wandering flows with finitely many singular points on compact surfaces are classified up to a graph-equivalence by using a topological invariant, called a Conley-Lyapunov-Peixoto graph, equipped with the rotation and the weight functions.…”

Section: Introductionmentioning

confidence: 99%

Graph representations of surface flows

Yokoyama¹

2017

Preprint

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We construct a complete invariant for non-wandering surface flows with finitely many singular points but without locally dense orbits. Precisely, we show that a flow v with finitely many singular points on a compact connected surface S is a non-wandering flow without locally dense orbits if and only if S/vex is a non-trivial embedded multi-graph, where the extended orbit space S/vex is the quotient space defined by x ∼ y if they belong to either a same orbit or a same multi-saddle connection. Moreover, collapsing edges of the non-trivial embedded multi-graph S/vex into singletons, the quotient space (S/vex)/ ∼ E is an abstract multi-graph with the Alexandroff topology with respect to the specialization order. Therefore the non-wandering flow v with finitely many singular points but without locally dense orbits can be reconstruct by finite combinatorial structures, which are the multi-saddle connection diagram and the abstract multi-graph (S/vex)/ ∼ E with labels. Moreover, though the set of topological equivalent classes of irrational rotations (i.e. minimal flows) on a torus is uncountable, the set of topological equivalent classes of non-wandering flows with finitely many singular points but without locally dense orbits on compact surfaces is enumerable by combinatorial structures algorithmically.

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Graphs and flows on surfaces

Cited by 13 publications

References 6 publications

A Polynomial Invariant of Graphs On Orientable Surfaces

A Polynomial Invariant of Graphs On Orientable Surfaces

Global Bifurcations in Generic One-Parameter Families with a Parabolic Cycle on S 2

Graph representations of surface flows

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