  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.
