Integer Representations of Convex Polygon Intersection Graphs

We determine tight bounds on the smallest-size integer grid needed to represent the $n$-node intersection graphs of a convex polygon $P$ with P given in rational coordinates. The intersection graphs use only polygons that are geometrically similar to $P$ (translates or homothets) and must be represented such that each corner of each polygon lies on a point of the grid. We show the following generic results: if $P$ is a parallelogram and only translates of $P$ are used, then an $\Omega(n^2) \times\Omega(n^2)$ grid is sufficient and is needed for some graphs; if $P$ is any other convex polygon and only translates of $P$ are used, then a $2^{\Omega(n)}\times 2^{\Omega(n)}$ grid is sufficient and is needed for some graphs; if $P$ is any convex polygon and arbitrary homothets of $P$ are allowed, then a $2^{\Omega(n)}\times 2^{\Omega(n)}$ grid is sufficient and is needed for some graphs. The results substantially improve earlier bounds and settle the complexity of representing convex polygon intersection graphs. The results also imply small polynomial certificates for the recognition problem for all graph classes considered.