Related papers: Construction and Classification of Combinatorial W…
In this work we present an algorithm with which any arbitrary cubic planar map may be constructed through successive edge insertion while simultaneously constructing a set of proper edge labels and Hamiltonian cycles for each configuration.…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
A mathematical model, describing some different weaving structures, is made in this article. The terms self-mirror and rotation-stable weaving structure are initiated here. There are used the properties and operations in the set of the…
Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic…
Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…
We present an algorithm to express Wilson lines that are defined on piecewise linear paths in function of their individual segments, reducing the number of diagrams needed to be calculated. The important step lies in the observation that…
When modeling microstructures, the computational resource requirements increase rapidly as the simulation domain becomes larger. As a result, simulating a small representative fraction under periodic boundary conditions is often a necessary…
The obvious way to construct a GDD (group-divisible design) recursively is to use Wilson's Fundamental Construction for GDDs (WFC). Then a PBD (pairwise balanced design) is often obtained by adding a new point to each group of the GDD.…
The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include…
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
Knitting, an ancient fiber art, creates a structured fabric consisting of loops or stitches. Publishing hand knitting patterns involves lengthy testing periods and numerous knitters. Modeling knitting patterns with graphs can help expedite…
We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial…
In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling…
Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled…
A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of…
Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function…
A weaving knot is an alternating knot whose minimal diagram is a closed braid of a lattice-like pattern. In this paper, the warping degree of a braid diagram is defined, and upper bounds of the unknotting number and the region unknotting…
A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The…
Two algorithms are presented for "compiling" influence diagrams into a set of simple decision rules. These decision rules define simple-to-execute, complete, consistent, and near-optimal decision procedures. These compilation algorithms can…