  Maximum parsimony is one of the most frequently-discussed tree reconstruction
methods in phylogenetic estimation. However, in recent years it has become more
and more apparent that phylogenetic trees are often not sufficient to describe
evolution accurately. For instance, processes like hybridization or lateral
gene transfer that are commonplace in many groups of organisms and result in
mosaic patterns of relationships cannot be represented by a single phylogenetic
tree. This is why phylogenetic networks, which can display such events, are
becoming of more and more interest in phylogenetic research. It is therefore
necessary to extend concepts like maximum parsimony from phylogenetic trees to
networks. Several suggestions for possible extensions can be found in recent
literature, for instance the softwired and the hardwired parsimony concepts. In
this paper, we analyze the so-called big parsimony problem under these two
concepts, i.e. we investigate maximum parsimonious networks and analyze their
properties. In particular, we show that finding a softwired maximum parsimony
network is possible in polynomial time. We also show that the set of maximum
parsimony networks for the hardwired definition always contains at least one
phylogenetic tree. Lastly, we investigate some parallels of parsimony to
different likelihood concepts on phylogenetic networks.
