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# On Disjoint Holes in Point Sets

This website provides programs to verify the computer-assisted results from my paper "On Disjoint Holes in Point Sets".

## Abstract of the paper

Given a set of points $S \subseteq \mathbb{R}^2$, a subset $X \subseteq S$, $|X|=k$, is called $k$-gon if all points of $X$ lie on the boundary of the convex hull $\mathrm{conv} (X)$, and $k$-hole if, in addition, no point of $S \setminus X$ lies in $\mathrm{conv} (X)$. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono and Urabe (2001). We also provide new bounds for three and more pairwise disjoint holes. Moreover, our program can be used to verify that every set of 17 points contains a 6-gon within significantly smaller computation time than the original program by Szekeres and Peters (2006).

## Short description of the programs

We provide a python program "h55.py" to formulate a SAT instance to verify that every set of 17 points contains two disjoint 5-holes. To execute the program, run
python h55.py 17

The SAT instance is written to a cnf-file, which then can be solved using a SAT solver such as glucose or picosat.
(The programs for interior-disjoint 5-holes and 6-gons are analogously.)

• The source code of the program "h55.py" to verify that every set of 17 points contains two disjoint 5-holes, i.e., $h(5,5)=17$. [download]
• The source code of the program "g6.py" to verify that every set of 17 points contains a 6-gon, i.e., $g(6)=17$. [download]