  In this work we are focused on the existence of Morse functions on a closed
manifold $M$ which are far from being ordered, i.e. whose Reeb graphs have
positive first Betti number, especially the maximal possible, equals
$\operatorname{corank}(\pi_1(M))$. In the case of $3$-manifolds we describe the
minimal number of critical points needed to construct such functions, which is
related with the number of vertices of degree $2$ in Reeb graphs. We define a
new invariant of $3$-manifold groups and their presentations, and using
Heegaard splittings we show its utility in determining occurrence of disordered
Morse functions. In particular, the Freiheitssatz, a result for one-relator
groups, allows us to calculate this invariant in the case of orientable
circle-bundles over a surface, which provides an interesting example of the
behaviour of Morse functions.
