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,, 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.

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 f1(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.

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. 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.

Structure of the Reeb graph defined by a smooth function

be a smooth function on a closed manifold.

The first Betti number of the Reeb graph

is one-dimensional, we consider only its first Betti number is the cycle rank of this graph. An upper bound holds

,

is the co-rank of the fundamental group, this bound is tight even in the class of simple Morse functions.

, for smooth functions this bound is also tight, and in terms of the genus

, for Morse functions, there is a better bound for the cycle rank. Since for Morse functions is a finite graph, the number of vertices with degree. Then

Leaf blocks of the Reeb graph

is a Morse or Morse–Bott function on a closed manifold has the structure of a finite graph. This finite graph has a specific structure, namely

• is Morse has no loopss, and all its leaf blocks are complete graphs, i.e., closed intervals

• is Morse–Bott has no loopss, and each its leaf block contains a vertex with degree at most 2

Description for Morse functions

is a Morse function with distinct critical values, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree is a saddle point splits into two.

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