Vindictive Voronoi Games and Stabbing Delaunay Circles

In this paper we consider the following problem: Given a set of $n$ Player1 sites in the plane and their Delaunay triangulation ${\cal D}$, place minimum possible Player2 sites such that in the resulting Delaunay triangulation ${\cal D}'$ of the sites of both Players, the neighborship between Player1 sites are as less as possible. We first consider placing minimum number of Player2 sites such that no two Player1 sites are neighbors in ${\cal D}'$. We show that to isolate a Player1 site $p$, two Player2 ites are both necessary and sufficient if $p$ is in the convex hull of ${\cal D}$, otherwise three Player2 sites are both necessary and sufficient. This gives a liner time algorithm to individually isolate all Player1 sites by 3n-h$ Player2 sites, where $h$ is the number of sites in the convex hull of ${\cal D}$. Then we give two more algorithms for this problem. The next algorithm runs also in linear time and uses $3(n-1)-h$ Player2 sites, but is much simpler. Our next algorithm uses $5|{\cal M}|$ sites, where $|{\cal M}|$ is the size of a maximum matching in ${\cal D}$, and runs in $O(\sqrt{n}m)$ time, where $m$ is the number of edges of ${\cal D}$. Then we consider isolating sites by components, where a component in ${\cal D'}$ is a maximal connected subset of sites of the same player. We show that it is possible to place $n$ Player2 sites such that in ${\cal D'}$ the number of components among Player1 sites is higher than that among Player2 sites. The above problems are related to at least two existing well known research topics: \emph{Voronoi games}, where the strategy of each of the two players is to place sites such that in the resulting Voronoi diagram some certain criteria is optimized for each player, and \emph{proximity graphs}, where this problem is known as \emph{minimum stabbing set of Delaunay circles}. Our bound of $5|{\cal M}|$ for the first problem would work better than known bound for the minimum stabbing set of Delaunay circles if ${\cal M}$ has a smaller size.