Reeb graph

A Reeb graph^[1] (named after Georges Reeb by René Thom) is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold.^[2] According to ^[3] a similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem.^[4] Proposed by G. Reeb as a tool in Morse theory,^[5] Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields ${\displaystyle \psi }$, ${\displaystyle \lambda }$, and ${\displaystyle \phi }$ arising from the conditions ${\displaystyle \nabla \psi =\lambda \nabla \phi }$ and ${\displaystyle \lambda \neq 0}$, because these relationships are single-valued when restricted to a region associated with an individual edge of the Reeb graph. This general principle was first used to study neutral surfaces in oceanography.^[6]

Reeb graph of the height function on the torus.

Reeb graphs have also found a wide variety of applications in computational geometry and computer graphics,^[1][7] including computer aided geometric design, topology-based shape matching,^[8][9][10] topological data analysis,^[11] topological simplification and cleaning, surface segmentation ^[12] and parametrization, efficient computation of level sets, neuroscience,^[13] and geometrical thermodynamics.^[3] In a special case of a function on a flat space (technically a simply connected domain), the Reeb graph forms a polytree and is also called a contour tree.^[14]

Level set graphs help statistical inference related to estimating probability density functions and regression functions, and they can be used in cluster analysis and function optimization, among other things. ^[15]

Formal definition

Given a topological space X and a continuous function f: X → R, define an equivalence relation ~ on X where p~q whenever p and q belong to the same connected component of a single level set f⁻¹(c) for some real c. The Reeb graph is the quotient space X /~ endowed with the quotient topology.

Generally, this quotient space does not have the structure of a finite graph. Even for a smooth function on a smooth manifold, the Reeb graph can be not one-dimensional and even non-Hausdorff space.^[16]

In fact, the compactness of the manifold is crucial: The Reeb graph of a smooth function on a closed manifold is a one-dimensional Peano continuum that is homotopy equivalent to a finite graph.^[16] In particular, the Reeb graph of a smooth function on a closed manifold with a finite number of critical values –which is the case of Morse functions, Morse–Bott functions or functions with isolated critical points – has the structure of a finite graph.^[17]

Structure of the Reeb graph defined by a smooth function

Let ${\displaystyle f:M\to {\mathbb {R} }}$ be a smooth function on a closed manifold ${\displaystyle M}$. The structure of the Reeb graph ${\displaystyle R_{f}}$ depends both on the manifold ${\displaystyle M}$ and on the class of the function ${\displaystyle f}$.

The first Betti number of the Reeb graph

Since for a smooth function on a closed manifold, the Reeb graph ${\displaystyle R_{f}}$ is one-dimensional,^[16] we consider only its first Betti number ${\displaystyle b_{1}(R_{f})}$; if ${\displaystyle R_{f}}$ has the structure of a finite graph, then ${\displaystyle b_{1}(R_{f})}$ is the cycle rank of this graph. An upper bound holds^[18][16]

${\displaystyle b_{1}(R_{f})\leq corank(\pi _{1}(M))}$,

where ${\displaystyle corank(\pi _{1}(M))}$ is the co-rank of the fundamental group of the manifold. If ${\displaystyle \dim M\geq 3}$, this bound is tight even in the class of simple Morse functions.^[19]

If ${\displaystyle \dim M=2}$, for smooth functions this bound is also tight, and in terms of the genus ${\displaystyle g}$ of the surface ${\displaystyle M^{2}}$ the bound can be rewritten as $${\displaystyle b_{1}(R_{f})\leq {\begin{cases}g,&{\text{if }}M^{2}{\text{ is orientable }}\\g/2,&{\text{if }}M^{2}{\text{ is non-orientable }}.\end{cases}}}$$

If ${\displaystyle \dim M=2}$, for Morse functions, there is a better bound for the cycle rank. Since for Morse functions, the Reeb graph ${\displaystyle R_{f}}$ is a finite graph,^[17] we denote by ${\displaystyle N_{2}}$ the number of vertices with degree 2 in ${\displaystyle R_{f}}$. Then^[20] $${\displaystyle b_{1}(R_{f})\leq {\begin{cases}g-N_{2},&{\text{if }}M^{2}{\text{ is orientable }}\\(g-N_{2})/2,&{\text{if }}M^{2}{\text{ is non-orientable }}.\end{cases}}}$$

Leaf blocks of the Reeb graph

If ${\displaystyle f:M\to R}$ is a Morse or Morse–Bott function on a closed manifold, then its Reeb graph ${\displaystyle R_{f}}$ has the structure of a finite graph.^[17] This finite graph has a specific structure, namely

• If ${\displaystyle f}$ is Morse, then ${\displaystyle R_{f}}$ has no loops, and all its leaf blocks are complete graphs ${\displaystyle K_{2}}$, i.e., closed intervals^[21]
• If ${\displaystyle f}$ is Morse–Bott, then ${\displaystyle R_{f}}$ has no loops, and each its leaf block contains a vertex with degree at most 2^[22]

Description for Morse functions

If ${\displaystyle f}$ is a Morse function with distinct critical values, the Reeb graph can be described more explicitly. Its nodes, or vertices, correspond to the critical level sets ${\displaystyle f^{-1}(c)}$. The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set ${\displaystyle f^{-1}(t)}$ as ${\displaystyle t}$ passes through the critical value ${\displaystyle c}$. For example, if ${\displaystyle c}$ is a minimum or a maximum of ${\displaystyle f}$, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1. If ${\displaystyle c}$ is a saddle point of index 1 and two components of ${\displaystyle f^{-1}(t)}$ merge at ${\displaystyle t=c}$ as ${\displaystyle t}$ increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y". The same reasoning applies if the index of ${\displaystyle c}$ is ${\displaystyle dimX-1}$ and a component of ${\displaystyle f^{-1}(c)}$ splits into two.