  A geometric graph $G$ is $xy-$monotone if each pair of vertices of $G$ is
connected by a $xy-$monotone path. We study the problem of producing the
$xy-$monotone spanning geometric graph of a point set $P$ that (i) has the
minimum cost, where the cost of a geometric graph is the sum of the Euclidean
lengths of its edges, and (ii) has the least number of edges, in the cases that
the Cartesian System $xy$ is specified or freely selected. Building upon
previous results, we easily obtain that the two solutions coincide when the
Cartesian System is specified and are both equal to the rectangle of influence
graph of $P$. The rectangle of influence graph of $P$ is the geometric graph
with vertex set $P$ such that two points $p,q \in P$ are adjacent if and only
if the rectangle with corners $p$ and $q$ does not include any other point of
$P$. When the Cartesian System can be freely chosen, we note that the two
solutions do not necessarily coincide, however we show that they can both be
obtained in $O(|P|^3)$ time. We also give a simple $2-$approximation algorithm
for the problem of computing the spanning geometric graph of a $k-$rooted point
set $P$, in which each root is connected to all the other points (including the
other roots) of $P$ by $y-$monotone paths, that has the minimum cost.
