Reeb graph

A Reeb graph (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. A similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem. Proposed by G. Reeb as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields \psi , \lambda , and \phi arising from the conditions \nabla \psi = \lambda \nabla \phi and \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. Reeb graphs have also found a wide variety of applications in computational geometry and computer graphics, including computer aided geometric design, topology-based shape matching, topological data analysis, topological simplification and cleaning, surface segmentation and parametrization, efficient computation of level sets, neuroscience, and geometrical thermodynamics. 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.…

A Reeb graph (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. A similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem. Proposed by G. Reeb as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields \psi , \lambda , and \phi arising from the conditions \nabla \psi = \lambda \nabla \phi and \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. Reeb graphs have also found a wide variety of applications in computational geometry and computer graphics, including computer aided geometric design, topology-based shape matching, topological data analysis, topological simplification and cleaning, surface segmentation and parametrization, efficient computation of level sets, neuroscience, and geometrical thermodynamics. 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. 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.

Structure of the Reeb graph defined by a smooth function

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

The first Betti number of the Reeb graph

Since for a smooth function on a closed manifold, the Reeb graph R_f is one-dimensional, we consider only its first Betti number b_1(R_f) ; if R_f has the structure of a finite graph, then b_1(R_f) is the cycle rank of this graph. An upper bound holds b_1(R_f)\le corank(\pi_1(M)) , where corank(\pi_1(M)) is the co-rank of the fundamental group of the manifold. If \dim M\ge3 , this bound is tight even in the class of simple Morse functions. If \dim M=2 , for smooth functions this bound is also tight, and in terms of the genus g of the surface M^2 the bound can be rewritten as b_1(R_f)\le \begin{cases} g, & \text{if } M^2 \text{ is orientable }\\ g/2, & \text{if } M^2 \text{ is non-orientable }. \end{cases} If \dim M=2 , for Morse functions, there is a better bound for the cycle rank. Since for Morse functions, the Reeb graph R_f is a finite graph, we denote by N_2 the number of vertices with degree 2 in R_f . Then b_1(R_f)\le \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 f:M\to R is a Morse or Morse–Bott function on a closed manifold, then its Reeb graph R_f has the structure of a finite graph. This finite graph has a specific structure, namely If f is Morse, then R_f has no loops, and all its leaf blocks are complete graphs K_2 , i.e., closed intervals If f is Morse–Bott, then R_f has no loops, and each its leaf block contains a vertex with degree at most 2 If 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 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 f^{-1}(t) as t passes through the critical value c . For example, if c is a minimum or a maximum of f , a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1. If c is a saddle point of index 1 and two components of f^{-1}(t) merge at t = c as 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 c is dim X-1 and a component of f^{-1}(c) splits into two.

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