Book embeddings of Reeb graphs
This work provides a theoretical advancement in computational topology for researchers in mathematics and data analysis, but it appears incremental as it builds on existing Reeb graph concepts.
The paper tackles the problem of embedding Reeb graphs, which represent topological features of simplicial complexes, into multi-page books, and results in a canonical embedding with a unique linear code.
Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding component of the level set $f^{-1}(t)$ changes. Then we can label every arc of $Reeb(f,X)$ with the Betti numbers $(β_1,β_2,\dots,β_d)$ of the corresponding $d$-dimensional component of a level set. The homology labels give more information about the original complex $X$ 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.