  Rooted phylogenetic networks are used to describe evolutionary histories that
contain non-treelike evolutionary events such as hybridization and horizontal
gene transfer. In some cases, such histories can be described by a phylogenetic
base-tree with additional linking arcs, which can for example represent gene
transfer events. Such phylogenetic networks are called tree-based. Here, we
consider two possible generalizations of this concept to nonbinary networks,
which we call tree-based and strictly-tree-based nonbinary phylogenetic
networks. We give simple graph-theoretic characterizations of tree-based and
strictly-tree-based nonbinary phylogenetic networks. Moreover, we show for each
of these two classes that it can be decided in polynomial time whether a given
network is contained in the class. Our approach also provides a new view on
tree-based binary phylogenetic networks. Finally, we discuss two examples of
nonbinary phylogenetic networks in biology and show how our results can be
applied to them.
