Title: Nearest-neighbour Markov point processes on graphs with Euclidean edges,
Abstract: We define nearest-neighbour point processes on graphs with Euclidean edges
and linear networks. They can be seen as the analogues of renewal processes on
the real line. We show that the Delaunay neighbourhood relation on a tree
satisfies the Baddeley--M{\o}ller consistency conditions and provide a
characterisation of Markov functions with respect to this relation. We show
that a modified relation defined in terms of the local geometry of the graph
satisfies the consistency conditions for all graphs with Euclidean edges.