  Recently, so-called treebased phylogenetic networks have gained considerable
interest in the literature, where a treebased network is a network that can be
constructed from a phylogenetic tree, called the base tree, by adding
additional edges. The main aim of this manuscript is to provide some sufficient
criteria for treebasedness by reducing phylogenetic networks to related graph
structures. While it is generally known that deciding whether a network is
treebased is NP-complete, one of these criteria, namely edgebasedness, can be
verified in linear time. Surprisingly, the class of edgebased networks is
closely related to a well-known family of graphs, namely the class of
generalized series parallel graphs, and we will explore this relationship in
full detail. Additionally, we introduce further classes of treebased networks
and analyze their relationships.
