Combinatorial Modifications of Reeb Graphs and the Realization Problem

We prove that, up to homeomorphism, any graph subject natural necessary conditions on orientation and the number of cycles can be realized as Reeb a Morse function given closed manifold $M$. Along way, we show that $\mathcal{R}(M)$, i.e. maximal among all graphs functions $M$, is equal corank fundamental group $\pi_1(M)$, thus extending previous result Gelbukh non-orientable case.