  We deal with Matveev complexity of compact orientable 3-manifolds represented
via Heegaard diagrams. This lead us to the definition of modified Heegaard
complexity of Heegaard diagrams and of manifolds. We define a class of
manifolds which are generalizations of Dunwoody manifolds, including cyclic
branched coverings of two-bridge knots and links, torus knots, some pretzel
knots, and some theta-graphs. Using modified Heegaard complexity, we obtain
upper bounds for their Matveev complexity, which linearly depend on the order
of the covering. Moreover, using homology arguments due to Matveev and Pervova
we obtain lower bounds.
