Reeb graph

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The Reeb graph is a topological graph named after the French mathematician Georges Reeb (1920–1993). It has its origin in the Morse theory .

definition

Let be a continuous scalar function over the compact manifold . The Reeb graph of with regard to the quotient space of the of the equivalence relation is induced. It applies if and only if ${\displaystyle f\colon M\rightarrow \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle x\sim y}$ ${\displaystyle x\sim y}$

• ${\displaystyle f(x)=f(y)}$ and
• ${\displaystyle x}$and in the same connected component of lying. ${\displaystyle y}$ ${\displaystyle f^{-1}(f(x))}$

That is, nodes arise at points where the topology of relative changes. These are also known as critical points. ${\displaystyle M}$ ${\displaystyle f}$