Efficient enumeration of at most k-out polygons

Let $S$ be a set of $n$ points in the Euclidean plane and general position i.e., no three points are collinear. An \emph{at most $k$-out polygon of $S$} is a simple polygon such that each vertex is a point in $S$ and there are at most $k$ points outside the polygon. In this paper, we consider the problem of enumerating all the at most $k$-out polygon of $S$. We propose a new enumeration algorithm for the at most $k$-out polygons of a point set. Our algorithm enumerates all the at most $k$-out polygons in $\mathcal{O}(n^2 \log{n})$ delay, while the running time of an existing algorithm is $\mathcal{O}(n^3 \log{n})$ delay.