computational geometry software developers

Abstract: String graphs are intersection graphs of continuous curves in the plane. In 2010, Fox and Pach proved that string graphs with $m$ edges admit balanced vertex separators of size $\mathcal{O}(m^{3/4}\sqrt{\log m})$ and conjectured that this bound can be improved to $\mathcal{O}(\sqrt{m})$, which would be optimal. After partial progress, Lee claimed in 2017 to prove this conjecture, and more generally to prove analogous separator theorems for region intersection graphs over minor-closed graph families. A key intermediate step in Lee's approach was a padded decomposition scheme for region intersection graphs. In 2024, Korhonen and Lokshtanov extended Lee's approach to the broader setting of graph classes excluding a fixed graph as an induced minor. These results have since found applications in a wide range of works. Later in 2024, a flaw was discovered in the construction of Lee's padded decomposition scheme, invalidating Lee's as well as Korhonen and Lokshtanov's separator results, and many works relying on them. In this work, we prove that string graphs with $m$ edges admit balanced separators of size $\mathcal{O}(\sqrt{m})$, and more generally obtain analogous $\mathcal{O}(\sqrt{m})$ bounds for region intersection graphs over fixed minor-closed graph families and for graphs excluding a fixed graph as an induced minor, thereby repairing the vast majority of the affected works. Furthermore, our balanced separators can be found in polynomial time. Our proof uses a different approach from Lee's and does not rely on padded decompositions. Since padded decompositions have also found applications independent of the separator theorems, particularly for string graphs, we also prove a padded decomposition result for region intersection graphs that is optimal for string graphs. We have formalized both the balanced separator and the padded decomposition results in Lean 4 with mathlib.

Abstract: Given a set $S$ of $n$ points in general position in the plane, we define the $C$-set $C(p,q)$ of an oriented edge $(p,q)$ as the set of extreme points of the visibility set $V(p,q)$ facing $(p,q)$. Since every point of $V(p,q)$ that is not in $C(p,q)$ gives rise to an empty convex pentagon having $(p,q)$ as a boundary edge, it is natural to study the properties of $C$-sets in order to bound the number of empty convex pentagons of $S$. We prove tight bounds on the intersection size and union complexity of $C$-sets sharing an endpoint. We also consider the problem of bounding the number of empty pentagons and hexagons in terms of the number of interior and boundary points of $S$. While the hexagon bound is weak, it offers a potential step toward a superlinear lower bound. On the other hand, the two-parameter bound for pentagons, combined with the structural results on $C$-sets, yields a simple proof of an $Ω(n^{3/2})$ bound for the number of pentagons. While this is a significant improvement over previously published bounds, a recent preprint \cite{Astudillo2026} attains a better bound using completely different methods. Thus, the primary contribution of this paper lies in the structural results about $C$-sets and the two-parameter bounds for empty pentagons and hexagons.

Abstract: We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.