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Combinatorial modifications of Reeb graphs and the realization problem
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abstract
We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold $M$. Along the way, we show that the Reeb number $\mathcal{R}(M)$, i.e. the maximum cycle rank among all Reeb graphs of functions on $M$, is equal to the corank of fundamental group $\pi_1(M)$, thus extending a previous result of Gelbukh to the non-orientable case.
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Cited by 1 Pith paper
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On Reeb graphs induced from smooth functions on closed or open manifolds
Any finite connected graph with edges labeled 0 or 1 and obeying simple balance rules is the Reeb graph of a smooth function on an orientable surface or higher-dimensional manifold, with level sets prescribed as circl...
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