Counting Blanks in Polygonal Arrangements

TL;DR

This paper investigates the problem of filling polygonal regions with non-overlapping tiles and blanks, providing tight bounds on the number of blanks needed across various geometric configurations.

Contribution

It offers the first comprehensive analysis and tight bounds for the number of blanks required in polygonal arrangements under multiple geometric constraints.

Findings

Tight bounds established for general polygons.

Results for convex figures and rectangles.

Bounds for rectilinear polygons with rectangular toppings.

Abstract

Inside a two dimensional region ("cake"), there are $m$ non-overlapping tiles of a certain kind ("toppings"). We want to expand the toppings while keeping them non-overlapping, and possibly add some blank pieces of the same "certain kind", such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axes-parallel rectangles. (4) The cake is an axes-parallel rectilinear polygon and the toppings are axes-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.