Book embeddings of Reeb graphs
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Let X be a simplicial complex with a piecewise linear function f:X→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,...,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.
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