  Given a set of points in the plane, a covering path is a polygonal path that
visits all the points. In this paper we consider covering paths of the vertices
of an n x m grid. We show that the minimal number of segments of such a path is
$2\min(n,m)-1$ except when we allow crossings and $n=m\ge 3$, in which case the
minimal number of segments of such a path is $2\min(n,m)-2$, i.e., in this case
we can save one segment. In fact we show that these are true even if we
consider covering trees instead of paths.
  These results extend previous works on axis-aligned covering paths of n x m
grids and complement the recent study of covering paths for points in general
position, in which case the problem becomes significantly harder and is still
open.
