Greedy walks on two lines

The greedy walk is a walk on a point process that always moves from its current position to the nearest not yet visited point. We consider here various point processes on two lines. We look first at the greedy walk on two independent one-dimensional Poisson processes placed on two intersecting lines and prove that the greedy walk almost surely does not visit all points. When a point process is defined on two parallel lines, the result depends on the definition of the process: If each line has a copy of the same realisation of a homogeneous Poisson point process, then the walk almost surely does not visit all points of the process. However, if each point of this process is removed with probability $p$ from either of the two lines, independently of the other points, then the walk almost surely visits all points. Moreover, the greedy walk on two parallel lines, where each line has a copy of the same realisation of a homogeneous Poisson point process, but one copy is shifted by some small $s$, almost surely visits all points.