  Semidirected networks have received interest in evolutionary biology as the
appropriate generalization of unrooted trees to networks, in which some but not
all edges are directed. Yet these networks lack proper theoretical study. We
define here a general class of semidirected phylogenetic networks, with a
stable set of leaves, tree nodes and hybrid nodes. We prove that for these
networks, if we locally choose the direction of one edge, then globally the set
of paths starting by this edge is stable across all choices to root the
network. We define an edge-based representation of semidirected phylogenetic
networks and use it to define a dissimilarity between networks, which can be
efficiently computed in near-quadratic time. Our dissimilarity extends the
widely-used Robinson-Foulds distance on both rooted trees and unrooted trees.
After generalizing the notion of tree-child networks to semidirected networks,
we prove that our edge-based dissimilarity is in fact a distance on the space
of tree-child semidirected phylogenetic networks.
