# Planar trees as complete topological invariants of Morse flows with a sink
on the 2-sphere

Oleksandr Pryshliak

###### Abstract

To investigate the topological structure of Morse flows with a sink on the
2-sphere we use the planar tree as complete topological invariant of the flow.
We give a list of all planar tree with at least 7 edges. We use a list of
rooted planar trees.

Key words and phrases. Morse flow, planar graphs, spherical graph, topological
invariant.

### Introduction

###  0.1 Relevance of the topic

The study of the topological structure of embedded trees on a plane is a
topical research topic for several reasons:

Applications in various scientific fields: Topological tree embeddings on a
plane have a wide range of applications in various scientific fields, such as
computer graphics, computer visualization, bioinformatics, social networks,
transportation networks, and many others. The study of topological structures
that arise when nesting trees on a plane can help to understand the
interactions between objects in different contexts and use this knowledge in
various scientific and practical applications.

Challenges and problems: Embedded trees on a plane is a non-trivial problem,
as it requires taking into account a number of restrictions, such as
restrictions on the intersection of edges, restrictions on the distance
between nodes, and others. Understanding the topological structure of nesting
trees on a plane can help solve complex problems and challenges related to the
development of efficient nesting algorithms, optimizing interactions between
objects, and improving data visualization.

Development of scientific methods: The study of the topological structure of
nesting trees on a plane contributes to the development of scientific methods
and approaches, such as graph theory, computer geometry, optimization methods,
and others.

###  0.2 Literature review

In the [56] definition basic concepts and properties of topological nesting of
trees on a plane, such as the concepts of "adjacency", "levels", "visibility"
and "external face". The authors developed the concept of "canonical
embeddings" and established a connection between them and the topological
properties of trees.

In [57], the problem of placing the vertices of trees on a plane in such a way
that the edges are straight segments was investigated. The concept of
"straight-line visible edges" was introduced and it was shown that not every
tree has direct attachments on the plane.

In [58], the authors studied the structural properties of nesting trees on a
plane, in particular, for the first time defined the concept of "planar
separators" (planar separators) and showed how to use them to construct
parallel nestings of trees on a plane.

In [59], the authors proposed a method for constructing "beautiful" nestings
of trees on a plane.

Book[61] provides a general overview of algorithms for graph visualization,
including techniques for placing tree vertices on a plane.

[62] can be considered a reference publication that contains articles from
leading researchers in the field of graph drawing and visualization. In it,
you can find sections devoted to the topological properties of nesting trees
on the plane and the corresponding algorithms.

In the collection of papers [63] presented at the Graph Drawing 2016
conference, the authors presented the latest research in the field of graph
drawing, including work on the topological structure of nesting trees on a
plane.

Work [64] is devoted to theory and algorithms for working with planar graphs,
including placing tree vertices on the plane.

In [60] the author offers various methods and models for placing graph
vertices on a plane.

The application of nested trees to study is especially interesting topological
properties of functions and dynamic systems – such properties that do not
depend on the choice of coordinate systems, measurement methods, etc. For
typical systems, they do not change with small changes in the parameters
defining this system.

Topological classification of flows on closed surfaces was obtained in [7, 19,
23, 24, 25, 43, 1, 32, 35, 48, 33, 34, 41, 31], and on surfaces with a
boundary y [5, 8, 21, 20, 35, 32, 28, 40, 25, 41, 44]. Complete topological
invariants of Morse-Smale flows on 3-manifolds was constructed in [49, 51, 39,
27, 29, 13, 4, 3].

Morse streams are gradient streams for Morse functions. If we fix the value of
the function at special points, then the structure of the flow determines the
structure of the function [22, 47]. Therefore, Morse–Smale flows
classification is related to the classification of the Morse functions.

Topological invariants of functions on oriented surfaces were constructed in
[18] and [45] and in [22] for unoriented surfaces, and in [10, 16, 17, 30] for
surfaces with a boundary, in [38] for non-compact surfaces.

Topological invariants of smooth functions were also studied in [6, 9, 17, 16,
38, 37, 42, 22, 27, 55, 50, 2, 46], for manifolds with limit in [14, 17, 15],
and on 3- and 4-dimensional manifolds in [36, 26].

For a first introduction to the topological theory of functions and dynamical
systems, we recommend [30, 52, 53, 54, 55].

###  0.3 The purpose of the paper

The main goal of this work is to describe all possible non-isomorphic planar
trees with no more than 8 vertices and use it for topological classifications
of Morse flows with one sink on $S^{2}$.

###  0.4 Research methods

To solve the main problem, we will use known algorithms for the classification
of planar-rooted trees. We will call the center of the tree the vertex that
has the greatest distance to the vertices of valence 1. The center of the tree
divides it into root trees. Our research method consists of gluing the roots
of root trees.

##  1 Planar-rooted trees

Plane-rooted trees are a particular type of tree in which the vertices are
located on a plane in such a way that the edges do not intersect. In these
trees, a special vertex is defined, known as the root of the tree, from which
all the edges emanate. Each vertex can have several child vertices that are
located lower on the plane than their parent vertices.

Figure 1: rooted trees with 0, 1, 2 and 3 edges

Planar-rooted trees are important data structures used in many fields such as
computer science, graphic design, data visualization, data processing
algorithms, and many others. They are one of the basic data structures for
representing a hierarchical organization of data, where one element (the root)
has a special status relative to other elements.

###  1.1 Rooted trees with 0, 1 and 2 edges

We will use the list of root trees given in the work [44]. In fig. 1 lists all
possible rooted trees with no more than 2 edges (1+1+2 trees).

###  1.2 Rooted trees with 3 edges

Figure 2: rooted trees with 3 edges

In fig. 2 all possible 5 rooted trees with 32 edges are given. Graphs 6 and 6a
are isomorphic on the plane, but not isomorphic on the half-plane (the root is
located on the boundary of the half-plane).

###  1.3 Rooted trees with 4 edges

Root trees with 4 edges are shown in Fig. 3.

Figure 3: rooted trees with 4 edges

Trees 4, 4a, 4b are isomorphic planar graphs, but not isomorphic rooted trees.

The same applies to pairs of graphs 11, 11a; 12, 12a; 15, 15a.

So, there are a total of 14 non-isomorphic rooted trees with 4 edges.

###  1.4 Rooted trees with 5 edges

Root trees with 5 edges are shown in Fig. 4,5. Only one root tree in each
plane tree class is given here.

Figure 4: rooted trees with 5 edges (part 1)

For tree 19 there are 4 non-isomorphic rooted trees, for 20 there are three
rooted trees, and for 21 there are also 3 rooted trees.

There are 3 non-isomorphic root trees for tree 22, 2 root trees for 23, 4 for
24, 2 for 25, and 2 root trees for 26.

Figure 5: rooted trees with 5 edges (part 2)

For tree 27 there are 2 non-isomorphic root trees, for 28 - 2 root trees, and
for 29 - one, for 30 - three, and for 31 - 2 root trees.

For tree 32, there are 2 non-isomorphic root trees, for 33 - one root tree,
and for 34 - one, for 35 - two, and for 36 and 37 one root tree each.

So, there are a total of 51 non-isomorphic rooted trees with 5 edges.

##  2 Planar trees

Since rooted trees are derived from planar trees, if you select a vertex on
them, planar trees with a small number of vertices can be specified by root
tree numbers.

###  2.1 Planar trees with no more than 6 vertices

We will get the following list:

  1. 1.

A planar tree with one vertex – 1.

  2. 2.

A planar tree with two vertices – 2.

  3. 3.

A planar tree with three vertices – 3=4.

  4. 4.

Two planar trees with four vertices – 5=7, 6=6a=8.

  5. 5.

Three planar trees with five vertices – 9=14, 10= 10a= 10b= 12= 12a= 15= 15a=
16, 11= 11a= 13= 17.

  6. 6.

Six planar trees with six vertices – 18=29, 19=23=30=34, 20=32, 21
=25=26=31=36, 22=24=33=35, 27=28=37.

###  2.2 Planar trees with 7 vertices

To obtain a list of planar trees, we will glue the root trees so that the
roots are glued to a common vertex. At the same time, it should be taken into
account that the same tree can be obtained by gluing different root trees. To
get rid of such ambiguity, we will assume that the resulting vertex is
central. By central we will understand the vertex that is at the greatest
distance from the vertices of valence 1 (there can be several central
vertices)

All possible non-isomorphic planar trees with 7 vertices are shown in fig.
6,7.

Figure 6: planar trees with 7 vertices (part 1)  Figure 7: planar trees with 7
vertices (part 2)

So there are 14 planar trees with 7 vertices.

###  2.3 Planar trees with 8 vertices

We will match and list the trees in the same way as in the previous section.

All possible non-isomorphic planar trees with 8 vertices are shown in fig.
8,9,10.

Figure 8: planar trees with 6 vertices (part 1)

Figure 9: planar trees with 6 vertices (part 2)  Figure 10: planar trees with
6 vertices (part 3)

So there are 26 planar trees with 8 vertices.

##  3 Morse flows on the 2-sphere

Consider a Morse flow with one sink on the sphere. Such flows are gradient
flows for the Morse function with one local maximum. Let us construct a graph
$G$, the vertices of which are the sources and the edges are the stable
manifolds of saddle points.

###### Theorem 1

The graph $G$ constructed in this way is a complete topological invariant of
the Morse flow. Moreover, the graph is a plane tree.

Proof. The fact that a graph is a complete topological invariant follows from
[19, 24, 25, 35]. Assume by contrast that the graph has a cycle. Then this
cycle splits the sphere into two regions. Each of these areas has at least one
drain. Then the total number of sinks is at least two, which contradicts the
conditions of the theorem. The resulting contradiction completes the proof.

Note that since the Euler characteristic of the sphere is equal to 2, then,
according to the Poincaré-Hopf theorem, the number of Morse flow sources with
one sink is one more than the number of saddles.

Consequence. On the 2-, there are such a number of topologically non-
equivalent Morse flows with one sink:

1 flow with one saddle,

1 flow with two saddles,

2 flows with three saddles,

3 flows with four saddles,

6 flows with five saddles,

14 flows with six saddles,

26 flows with seven saddles.

## Conclusion

In this paper, we have constructed a complete list of planar graphs with no
more than 8 vertices. When constructing, we used rooted trees and checked them
for isomorphism as planar graphs.

## References

  * [1] O. Akchurin, S. Bilun, and A. Prishlyak.  Three-color graph as the 1-skeleton of the 2-sphere triangulation.  arXiv preprint arXiv:2209.05737, 2022. doi:10.48550/ARXIV.2209.05737.
  * [2] S. Bilun and A. Prishlyak.  The closed morse 1-forms on closed surfaces.  Visn., Mat. Mekh., Kyv. Univ. Im. Tarasa Shevchenka, 2002(8):77–81, 2002.
  * [3] S. Bilun and A. Prishlyak.  Visualization of morse flow with two saddles on 3-sphere diagrams.  arXiv preprint arXiv:2209.12174, 2022. doi:10.48550/ARXIV.2209.12174.
  * [4] S. Bilun, A. Prishlyak, and A. Prus.  Morse flows with fixed points on the boundary of 3-manifold.  arXiv preprint arXiv:2209.04019, 2022. doi:10.48550/arXiv.2209.04019.
  * [5] S. Bilun, M. Hrechko, O. Myshnova, A. Prishlyak.  Structures of optimal discrete gradient vector fields on surface with one or two critical cells  arXiv preprint arXiv:2303.07258, 2023. doi:10.48550/arXiv.2303.07258.
  * [6] S. Bilun, A. Prishlyak, S. Stas, A. Vlasenko.  Topological structure of Morse functions on the projective plane  arXiv preprint arXiv:2303.03850, 2023. doi:10.48550/arXiv.2303.03850.
  * [7] S. Bilun, B. Hladysh, A. Prishlyak, V. Sinitsyn.  Gradient vector fields of codimension one on the 2-sphere with at most ten singular points  arXiv preprint arXiv:2303.10929, 2023. doi:10.48550/arXiv.2303.10929.
  * [8] S. Bilun, M. Loseva, O. Myshnova, A. Prishlyak.  Typical one-parameter bifurcations of gradient flows with at most six singular points on the 2-sphere with holes  arXiv preprint arXiv:2303.14975, 2023. doi:10.48550/arXiv.2303.14975.
  * [9] S. Bilun, B. Hladysh, A. Prishlyak, M. Roman.  Morse functions with four critical points on immersed 2-spheres.  arXiv preprint arXiv:2304.04392, 2023. doi:10.48550/arXiv.2304.04392.
  * [10] A.V. Bolsinov and A.T. Fomenko.  Integrable Hamiltonian systems. Geometry, Topology, Classification.  A CRC Press Company, Boca Raton London New York Washington, D.C., 2004\.  724 p.
  * [11] M. Borodzik, A. Némethi and A. Ranicki  Morse theory for manifolds with boundary.  Algebraic Geometric Topology, 16(2):971–1023, 2016. doi:10.2140/agt.2016.16.971.
  * [12] M. Goresky and R. MacPherson.  Stratified Morse Theory.  Springer, 1988. doi:10.1007/978-3-642-71714-7.
  * [13] C. Hatamian and A. Prishlyak.  Heegaard diagrams and optimal morse flows on non-orientable 3-manifolds of genus 1 and genus 2.  Proceedings of the International Geometry Center, 13(3):33–48, 2020\. doi:10.15673/tmgc.v13i3.1779.
  * [14] B. I. Hladysh and A. O. Pryshlyak.  Functions with nondegenerate critical points on the boundary of the surface.  Ukrainian Mathematical Journal, 68(1):29–41, 2016. doi:10.1007/s11253-016-1206-5.
  * [15] B. I. Hladysh and A. O. Pryshlyak.  Deformations in the general position of the optimal functions on oriented surfaces with boundary.  Ukrainian Mathematical Journal, 71(8):1173–1185, 2020. doi:10.1007/s11253-019-01706-8.
  * [16] B.I. Hladysh and A.O. Prishlyak.  Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold.  SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 13:050, 2017. doi:0.3842/SIGMA.2017.050.
  * [17] B.I. Hladysh and A.O. Prishlyak.  Simple morse functions on an oriented surface with boundary.  Журнал математической физики, анализа, геометрии, 15(3):354–368, 2019. doi:10.15407/mag15.03.354.
  * [18] A.S. Kronrod.  Functions of two variables.  Russian Mathematical Surveys, 5:24–134, 1950.
  * [19] Z. Kybalko, A. Prishlyak, and R. Shchurko.  Trajectory equivalence of optimal Morse flows on closed surfaces.  Proc. Int. Geom. Cent., 11(1):12–26, 2018. doi:10.15673/tmgc.v11i1.916.
  * [20] M. Loseva, A. Prishlyak.  Topological structure of optimal flows on the Girl’s surface.  Proceedings of the International Geometry Center, 15(3-4):184–202, 2022\. doi:10.15673/tmgc.v15i3-4.2338.
  * [21] M. Losieva and A. Prishlyak.  Topology of morse–smale flows with singularities on the boundary of a two-dimensional disk.  Pr. Mizhnar. Heometr. Tsentr, 9(2):32–41, 2016. doi:10.15673/tmgc.v9i2.279.
  * [22] D.P. Lychak and A.O. Prishlyak.  Morse functions and flows on nonorientable surfaces.  Methods of Functional Analysis and Topology, 15(03):251–258, 2009\.
  * [23] A.A. Oshemkov and V.V. Sharko.  Classication of morse-smale flows on two-dimensional manifolds.  Matem. Sbornik, 189(8):93–140, 1998.
  * [24] M.M. Peixoto.  On the classication of flows of 2-manifolds.  Dynamical Systems (Proc. Symp. Univ. of Bahia, Salvador, Brasil, 1971), pages 389–419, 1973.
  * [25] A.O. Prishlyak.  On graphs embedded in a surface.  Russian Mathematical Surveys, 52(4):844, 1997. doi:10.1070/RM1997v052n04ABEH002074.
  * [26] A.O. Prishlyak.  Conjugacy of Morse functions on 4-manifolds.  Russian Mathematical Surveys, 56(1):170, 2001.
  * [27] A.O. Prishlyak.  Morse–smale vector fields without closed trajectories on-manifolds.  Mathematical Notes, 71(1-2):230–235, 2002. doi:10.1023/A:1013963315626.
  * [28] A.O. Prishlyak.  On sum of indices of flow with isolated fixed points on a stratified sets.  Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 10(1):106–115, 2003.
  * [29] A.O. Prishlyak.  Complete topological invariants of morse-smale flows and handle decompositions of 3-manifolds.  Journal of Mathematical Sciences, 144:4492–4499, 2007.
  * [30] A.O. Prishlyak.  Topological properties of functions on two and three dimensional manifolds.  Palmarium Academic Pablishing, 2012.
  * [31] A. Prishlyak and L. Di Beo.  Flows with minimal number of singularities on the Boy’s surface  Proceedings of the International Geometry Center, 15(1):32–49, 2020\.
  * [32] A. Prishlyak and M. Loseva.  Topological structure of optimal flows on the girl’s surface.  Proceedings of the International Geometry Center, 15(3-4):184–202, 2022.
  * [33] A. Prishlyak, A. Prus, and S. Huraka.  Flows with collective dynamics on a sphere.  Proc. Int. Geom. Cent, 14(1):61–80, 2021. doi:10.15673/tmgc.v14i1.1902.
  * [34] A. Prishlyak and M. Loseva.  Topology of optimal flows with collective dynamics on closed orientable surfaces.  Proceedings of the International Geometry Center, 13(2):50–67, 2020\. doi:10.15673/tmgc.v13i2.1731.
  * [35] A. Prishlyak and A. Prus.  Morse-smale flows on torus with hole.  Proc. Int. Geom. Cent., 10(1):47–58, 2017. doi:10.15673/tmgc.v1i10.549.
  * [36] A.O. Prishlyak.  Equivalence of morse function on 3-manifolds.  Methods of Func. Ann. and Topology, 5(3):49–53, 1999.
  * [37] A.O. Prishlyak.  Conjugacy of morse functions on surfaces with values on a straight line and circle.  Ukrainian Mathematical Journal, 52(10):1623–1627, 2000. doi:10.1023/A:1010461319703.
  * [38] A.O. Prishlyak.  Morse functions with finite number of singularities on a plane.  Meth. Funct. Anal. Topol, 8:75–78, 2002.
  * [39] A.O. Prishlyak.  Topological equivalence of morse–smale vector fields with beh2 on three-dimensional manifolds.  Ukrainian Mathematical Journal, 54(4):603–612, 2002.
  * [40] A.O. Prishlyak.  Topological classification of m-fields on two-and three-dimensional manifolds with boundary.  Ukrainian Mathematical Journal, 55(6):966–973, 2003.
  * [41] A.O. Prishlyak and M.V. Loseva.  Optimal morse–smale flows with singularities on the boundary of a surface.  Journal of Mathematical Sciences, 243:279–286, 2019.
  * [42] A.O. Prishlyak and K.I. Mischenko.  Classification of noncompact surfaces with boundary.  Methods of Functional Analysis and Topology, 13(01):62–66, 2007\.
  * [43] A.O. Prishlyak and A.A. Prus.  Three-color graph of the morse flow on a compact surface with boundary.  Journal of Mathematical Sciences, 249(4):661–672, 2020. doi:10.1007/s10958-020-04964-1.
  * [44] A. Prishlyak and S. Stas.  Structures of the flows with a unique singular point on the 2-dimensional disk.  arXiv preprint arXiv:2304.00751, 2023. doi:10.48550/ARXIV.2304.00751.
  * [45] G. Reeb.  Sur les points singuliers d’une forme de pfaff complétement intégrable ou d’une fonction numérique.  C.R.A.S. Paris, 222:847—849, 1946.
  * [46] V.V. Sharko.  Functions on manifolds. Algebraic and topological aspects., volume 131 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [47] S. Smale.  On gradient dynamical systems.  Ann. of Math., 74:199–206, 1961.
  * [48] В.М. Кузаконь, В.Ф. Кириченко, and О.О. Пришляк.  Гладкi многовиди. Геометричнi та топологiчнi аспекти.  Працi Iнcтитуту математики НАН України.—2013.—97.—500 с, 2013.
  * [49] А.О. Пришляк.  Векторные поля Морса–Смейла с конечным числом особых траекторий на трехмерных многообразиях.  Доповiдi НАН України, (6):43–47, 1998.
  * [50] А.О. Пришляк.  Сопряженность функций Морса.  Некоторые вопросы совр. математики. Институт математики АН Украины, Киев, 1998.
  * [51] А.О. Пришляк.  Топологическая эквивалентность функций и векторных полей Морса—Смейла на трёхмерных многообразиях.  Топология и геометрия. Труды Украинского мат. конгресса, pages 29–38, 2001.
  * [52] О.О. Пришляк.  Теорія Морса.  Київський університет, 2002.
  * [53] О.О. Пришляк.  Диференціальна геометрія.  Київський університет, 2004.
  * [54] О.О. Пришляк.  Основи сучасної топології.  Київський університет, 2006.
  * [55] О.О. Пришляк.  Топологія многовидів.  Київський університет, 2015.
  * [56] Maurice Fraysseix, Pavol Pach, Richard Pollack  Planar Embeddings  Journal of Combinatorial Theory, Series B, 45(3):105-126, 1990.
  * [57] Hsueh-I Lu Chen  Straight-Line Drawings on Plane Graphs  SIAM Journal on Computing, 20(6):1048-1066, 1991.
  * [58] Gary L. Miller, Joseph Naor  Planar Separators and Parallel Polygon Triangulation  SIAM Journal on Computing, 20(3):449-456, 1991.
  * [59] Christoph Buchheim, Michael Jünger, Sebastian Leipert  Drawing Trees Nicely with Linear Area Requirements  Theoretical Computer Science, 281(1-2):21-48, 2002.
  * [60] Sergey G. Kobourov  Drawing Graphs: Methods and Models  CRC Press, Boca Raton, FL, USA, 2013. 498 pages.
  * [61] Giuseppe Di Battista, Peter Eades, Roberto Tamassia, Ioannis G. Tollis  Graph Drawing: Algorithms for the Visualization of Graphs  Prentice Hall, Upper Saddle River, NJ, USA, 1999. 384 pages.
  * [62] Roberto Tamassia  Handbook of Graph Drawing and Visualization  CRC Press, Boca Raton, FL, USA, 2013. 862 pages.
  * [63] Arikushi, Y., Speckmann, B. (eds.) (2017). Graph Drawing: 24th International Symposium, GD 2016, Athens, Greece, September 19-21, 2016, Revised Selected Papers. Springer. Cham, Switzerland.
  * [64] Pach, J., Agarwal, P.K. (2001). Planar Graphs: Theory and Algorithms. Cambridge University Press. Cambridge, UK.
  * [65] Tamassia, R., Di Battista, G. (1987). On Embedding a Graph in the Grid with the Minimum Number of Bends. SIAM Journal on Computing, 16(3), 421-444.