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 also present a polynomial algorithm for this question when n\leq 2k-c, conditionally on a conjecture of R.E. Jamison. For n=k, an algorithm in O(n^4) was proposed by Wade and Chu. In this case, our algorithm is linear and does not rely on Jamison's conjecture.
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 O(n^4) was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.
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