Let @math be a simplicial complex with a piecewise linear function @math . The Reeb graph @math is the quotient of @math , where we collapse each connected component of @math to a single point. Let the nodes of @math be all homologically critical points where any homology of the corresponding component of the level set @math changes. Then we can label every arc of @math with the Betti numbers @math of the corresponding @math -dimensional component of a level set. The homology labels give more information about the original complex @math than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.
@cite_9: Reeb graphs are compact shape descriptors which convey topological information related to the level sets of a function defined on the shape. Their definition dates back to 1946, and finds its root in Morse theory. Reeb graphs as shape descriptors have been proposed to solve different problems arising in Computer Graphics, and nowadays they play a fundamental role in the field of computational topology for shape analysis. This paper provides an overview of the mathematical properties of Reeb graphs and reconstructs its history in the Computer Graphics context, with an eye towards directions of future research.
Reeb graphs are used in image analysis @cite_9 as simple representations of shapes. The Reeb graph is an abstract graph that describes the linking structure of connected components in level sets of a real-valued function on a complex. Briefly, a complex is a union of vertices, edges, triangles, tetrahedra and so on. Formal Definition of a Reeb graph in section doesn't specify any canonical way to visualize this abstract graph in a low dimensional space.