Related papers: Construction and Classification of Combinatorial W…

The basic (and traditional) crossing number problem is to determine the minimum number of crossings in a topological drawing of an input graph in the plane. We develop a unified framework yielding fixed-parameter tractable (FPT) algorithms…

We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global…

We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge to edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles,…

We propose a simple new combinatorial model to study spaces of acyclic Jacobi diagrams, in which they are identified with algebras of words modulo operations. This provides a starting point for a word-problem type combinatorial…

Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which…

This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…

We develop a graphical calculus of manifold diagrams which generalises string and surface diagrams to arbitrary dimensions. Manifold diagrams are pasting diagrams for $(\infty, n)$-categories that admit a semi-strict composition operation…

We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of…

In this paper we extend counting of traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be done in linear time with…

In the article "Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane" we gave the construction of a discrete hull for a combinatorial pentagonal tiling of the plane. In this paper, we give the…

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines…

The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs.…

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand,…

One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…

Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for…

The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by…

We propose a graphical language that accommodates two monoidal structures: a multiplicative one for pairing and an additional one for branching. In this colored PROP, whether wires in parallel are linked through the multiplicative structure…

Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…