Abstract

Given a set $A$ of $n$ points (vertices) in general position in the plane, the complete geometric graph $K_n[A]$ consists of all $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3∕2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ randomly selected points, uniformly distributed in ${[0,1]}^2$, with probability tending to $1$ as $n\to \infty$, the edge set of $K_n[A]$ can be covered by $O(n\log <!--FUNCTION\; APPLICATION-->n)$ crossing-free paths and by $O(n\sqrt{\log <!--FUNCTION\; APPLICATION-->n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

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