Uniform 2D-Monotone Minimum Spanning Graphs

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.