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Francis Lazarus

Publications and source records attributed to Francis Lazarus.

10 recordsLinked to original sources
arXiv2025

On the Computation of Schrijver's Kernels

The geometry of a graph $G$ embedded on a closed oriented surface $S$ can be probed by counting the intersections of $G$ with closed curves on $S$. Of special interest is the map $c \mapsto \mu_G(c)$ counting the minimum number of intersections between $G$ and any curve freely homotopic to a given curve $c$. Schrijver [On the uniqueness of kernels, 1992] calls $G$ a kernel if for any proper graph minor $H$ of $G$ we have $\mu_H < \mu_G$. Hence, $G$ admits a minor $H$ which is a kernel and such that $\mu_G = \mu_H$. We show how to compute such a minor kernel of $G$ in $O(n^3 \log n)$ time where $n$ is the number of edges of $G$, and $g\ge 2$ is the genus of $S$. Our algorithm leverages a tight bound on the size of minimal bigons in a system of closed curves. It also relies on several subroutines of independent interest including the computation of the area enclosed by a curve and a test of simplicity for the lift of a curve in the universal covering of $S$. As a consequence of our minor kernel algorithm and a recent result of Dubois [Making multicurves cross minimally on surfaces, 2024], after a preprocessing that takes $O(n^3 \log n)$ time and $O(n)$ space, we are able to compute $\mu_G(c)$ in $O(g (n + \ell) \log(n + \ell))$ time given any closed walk $c$ with $\ell$ edges. The state-of-the-art algorithm by Colin de Verdi\`ere and Erickson [Tightening non-simple paths and cycles on surfaces, 2010] would avoid constructing a kernel but would lead to a computation of $\mu_G(c)$ in $O(g n \ell \log(n \ell))$ time (with a preprocessing that takes $O(gn\log n)$ time and $O(gn)$ space). Another consequence of the computation of minor kernels is the ability to decide in polynomial time whether two graph minors $H$ and $H'$ of $G$ satisfy $\mu_H = \mu_{H'}$.

cs.CG↗
arXiv2024

Computing the second and third systoles of a combinatorial surface

Given a weighted, undirected graph $G$ cellularly embedded on a topological surface $S$, we describe algorithms to compute the second shortest and third shortest closed walks of $G$ that are neither homotopically trivial in $S$ nor homotopic to the shortest non-trivial closed walk or to each other. Our algorithms run in $O(n^2\log n)$ time for the second shortest walk and in $O(n^3)$ time for the third shortest walk. We also show how to reduce the running time for the second shortest homotopically non-trivial closed walk to $O(n\log n)$ when both the genus and the number of boundaries are fixed. Our algorithms rely on a careful analysis of the configurations of the first three shortest homotopically non-trivial curves in $S$. As an intermediate step, we also describe how to compute a shortest essential arc between \emph{one} pair of vertices or between \emph{all} pairs of vertices of a given boundary component of $S$ in $O(n^2)$ time or $O(n^3)$ time, respectively.

cs.CG↗
arXiv2023

The Hyperbolic Plane in $\mathbb{E}^3$

We build an explicit $C^1$ isometric embedding $f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$, our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. To understand the behavior of $df_\infty$ we introduce a $formal$ $corrugation$ $process$ leading to a $formal$ $analogue$ $Φ_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure for $Φ_{\infty}$. We next prove that $df_\infty$ is close to $Φ_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbolν_{\infty}^Φ$ and $\boldsymbolν_{\infty}$, of respectively $Φ_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $Φ_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbolν_{\infty}$ towards $\boldsymbolν_{\infty}^Φ$ over circles of rational radii.

math.DG↗
arXiv2023

Algorithms for Length Spectra of Combinatorial Tori

Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order. In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity $n$ in time $O(n^2 \log \log n)$ so that, given a cycle with $\ell$ edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in $O(\ell+\log n)$ time. Moreover, given any positive integer $k$, the first $k$ values of its unmarked length spectrum can be computed in time $O(k \log n)$. Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.

cs.CG↗
arXiv2022

A Universal Triangulation for Flat Tori

A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially non-constructive, to produce PL isometric embeddings of flat tori. Our implementation produces embeddings with a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller (2000) and on recent works by Arnoux et al. (2021), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.

cs.CG↗
arXiv2020

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve $γ$, and a collection of disjoint normal curves $Δ$, there is a polynomial-time algorithm to decide if $γ$ lies in the normal subgroup generated by components of $Δ$ in the fundamental group of the surface after attaching the curves to a basepoint.

cs.CG↗
arXiv2019

Computing the Geometric Intersection Number of Curves

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algorithms to (1) compute the geometric intersection number of $c$ in $O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+\ell^4)$ time, (3) decide if the geometric intersection number of $c$ is zero, i.e. if $c$ is homotopic to a simple curve, in $O(n+\ell\log\ell)$ time. The algorithms for (2) and (3) are restricted to orientable surfaces, but the algorithm for (1) is also valid on non-orientable surfaces. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a $O(n+g^2\ell^2)$ time complexity on a genus $g$ surface without boundary. No polynomial time algorithm was known for problem (2) for surfaces without boundary. Interestingly, our solution to problem (3) provides a quasi-linear algorithm to a problem raised by Poincaré more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most $\ell$ in $O(n+ \ell^2)$ time.

cs.CG↗
arXiv2015

Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $Σ$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $Σ$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.

cs.CG↗
arXiv2011

On the homotopy test on surfaces

Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, but allow non-orientable surfaces when the basepoint is fixed. After O(|G|) time preprocessing independent of c and d, our algorithms answer the homotopy test in O(|c|+|d|) time, where |G|, |c| and |d| are the respective numbers of edges of G, c and d. As a byproduct we obtain linear time algorithms for the word problem and the conjugacy problem in surface groups. We present a geometric approach based on previous works by Colin de Verdière and Erickson.

cs.CG↗