Discrete & Computational Geometry最新文献

We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained for a fixed number of crossings. A random meander diagram is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. We use this for exploring geometric properties of a link complement. Specifically, we give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger’s algorithm.

Given a graph (G=(V,E)) and a set (S subseteq left( {begin{array}{c}V 2end{array}}right) ) of terminal pairs, the minimum multicut problem asks for a minimum edge set (delta subseteq E) such that there is no s-t-path in (G -delta ) for any ({s,t}in S). For (|S|=1) this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope (textsc {MultC}^square (G,S)) is the convex hull of all multicuts in G; the multicut dominant is given by (textsc {MultC}(G,S)=textsc {MultC}^square (G,S)+mathbb {R}^E_{{ge 0}}). The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.

We introduce the multi-width of a lattice polytope and use this to classify and count all lattice tetrahedra with multi-width ((1,w_2,w_3)). The approach used in this classification can be extended into a computer algorithm to classify lattice tetrahedra of any given multi-width. We use this to classify tetrahedra with multi-width ((2,w_2,w_3)) for small (w_2) and (w_3) and make conjectures about the function counting lattice tetrahedra of any multi-width.

A set ({mathcal {G}}) of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph (G in {mathcal {G}}) admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most ((3+o(1))log _2(n)) planar graphs on n vertices. This constitutes a double-exponential improvement over the previously best known bound of (O(ncdot 4^{n/11})) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer (nin [107,193]) there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size (O(log n)) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than (n^6) planar n-vertex graphs with no simultaneous embedding.

Normal complexes are orthogonal truncations of simplicial fans. In this paper, we develop the study of mixed volumes for normal complexes. Our main result is a sufficiency condition that ensures when the mixed volumes of normal complexes associated to a given fan satisfy the Alexandrov–Fenchel inequalities. By specializing to Bergman fans of matroids, we give a new proof of the Heron–Rota–Welsh Conjecture as a consequence of the Alexandrov–Fenchel inequalities for normal complexes.

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about (tfrac{4}{pi }). We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least ((log n)^{1/4 - o(1)}) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound (Omega (log ^{1/8}n)) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least ((log n)^{1 -o(1)}).

Given a set of points (P) and a set of regions (mathcal {O}), an incidence is a pair ((p,mathcalligra {o}) in Ptimes mathcal {O}) such that (pin mathcalligra {o}). We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of (Obigl ( k n(log n/log log n)^{d-1} bigr )) on the number of incidences between n points and n axis-parallel boxes in (mathbb {R}^d), if no k boxes contain k common points, that is, if the incidence graph between the points and the boxes does not contain (K_{k,k}) as a subgraph. This new bound improves over previous work, by Basit et al. (Forum Math Sigma 9:59, 2021), by more than a factor of (log ^d n) for (d >2). Furthermore, it matches a lower bound implied by the work of Chazelle (J ACM 37(2):200–212, 1990), for (k=2), thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.

For a closed Riemannian manifold (mathcal {M}) and a metric space S with a small Gromov–Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale (beta >0) at which the Vietoris–Rips complex of S is homotopy equivalent to (mathcal {M}). Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale (beta ) in order to provide sampling conditions for S to be homotopy equivalent to (mathcal {M}). In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of (beta ) using the sectional curvatures and convexity radius of (mathcal {M}) as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris–Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris–Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.