11.1 Definition and construction of Reeb graphs

Reeb graphs capture the connectivity of level sets in Morse functions on manifolds. They're built by identifying points in the same connected component of a level set, creating a graph structure that shows how these components evolve.

Nodes in Reeb graphs represent critical points where level set topology changes, while edges show connectivity between them. This structure helps track how connected components appear, disappear, split, or merge as the Morse function value changes.

Reeb Graph Definition

Morse Functions and Level Sets

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• A Reeb graph is a topological structure that captures the connectivity of level sets of a Morse function on a manifold
• Morse functions are real-valued smooth functions on a manifold with non-degenerate critical points
  • Non-degenerate critical points have a non-singular Hessian matrix
• Level sets are the preimages of a real value under the Morse function
  • Consist of all points on the manifold that map to the same value
  • Can be thought of as the "slices" of the manifold at different heights (for a height function)

Quotient Space Construction

• The Reeb graph is obtained as a quotient space of the manifold
  • Points in the same connected component of a level set are identified as equivalent
  • The resulting space is a graph structure
• Equivalent points are "glued together" to form the nodes and edges of the Reeb graph
  • Each connected component of a level set becomes a point (node) in the Reeb graph
  • Edges in the Reeb graph represent the evolution of connected components between critical points

Reeb Graph Components

Graph Nodes and Edges

• Nodes in the Reeb graph correspond to critical points of the Morse function
  • Minimum, maximum, and saddle points become nodes
  • At these points, the topology of the level sets changes (components appear, disappear, split, or merge)
• Edges in the Reeb graph represent the connectivity between critical points
  • An edge connects two nodes if there is a continuous path between the corresponding critical points that stays within the same connected component of the level sets

Connected Components and Critical Points

• The Reeb graph tracks the evolution of connected components of level sets
  • As the value of the Morse function changes, the number and connectivity of components can change
• Critical points mark the values where the topology of the level sets changes
  • At a minimum, a new connected component appears (birth)
  • At a maximum, a connected component disappears (death)
  • At a saddle point, connected components can split or merge (bifurcation)

Related Structures

Contour Trees

• A contour tree is a special case of a Reeb graph for scalar functions on simply connected domains
  • Simply connected means the domain has no holes or voids
• In a contour tree, each connected component of a level set becomes a point, and points are connected if the components are nested
  • Results in a tree structure rather than a general graph
• Contour trees are used in visualization and analysis of scalar fields
  • Provide a compact representation of the topology of level sets
  • Allow for efficient queries and simplification of the scalar field