NP-completeness of slope-constrained drawing of complete graphs

Abstract

We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $k\geq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $n\leq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $\mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.