  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.
