Drawing the Almost Convex Set in an Integer Grid of Minimum Size

In 2001, K\'arolyi, Pach and T\'oth introduced a family of point sets to solve an Erd\H{o}s-Szekeres type problem; which have been used to solve several other Ed\H{o}s-Szekeres type problems. In this paper we refer to these sets as nested almost convex sets. A nested almost convex set $\mathcal{X}$ has the property that the interior of every triangle determined by three points in the same convex layer of $\mathcal{X}$, contains exactly one point of $\mathcal{X}$. In this paper, we introduce a characterization of nested almost convex sets. Our characterization implies that there exists at most one (up to order type) nested almost convex set of $n$ points. We use our characterization to obtain a linear time algorithm to construct nested almost convex sets of $n$ points, with integer coordinates of absolute values at most $O(n^{\log_2 5})$. Finally, we use our characterization to obtain an $O(n\log n)$-time algorithm to determine whether a set of points is a nested almost convex set.