Non-crossing Graphs Research Articles

Enumeration of bipartite non-crossing geometric graphs

By using the Riordan array method together with combinatorial species theory, we study enumeration problems for geometric graphs which are connected bipartite non-crossing graphs with n + 1 points in convex position. It is performed from two different points of view, one is by the distance between two vertices 1 and n + 1 , and the other one is by the degree of the vertex n + 1 . As a result, we obtain a production matrix for such geometric graphs, and a formula for the number of connected bipartite graphs, which gives an answer to an open question about the connected bipartite non-crossing graphs.

Method of coordinating joint operation of air starter and auxiliary power unit and determining the gas turbine engine starting time

The article describes the method developed by the authors to coordinate the working process of the auxiliary power plant (APP) and the starter air turbine (SAT) used for starting a gas turbine engine (GTD). This method is used to test the possibility of joint operation of the APP and the air turbine in the gas turbine engine starting system in given operating modes. The method is based on combining the APP and turbine characteristics shown in the same coordinates on the same field and checking for intersection points. The condition of joint operation is fulfilled in them. Non-crossing graphs indicate the impossibility of joint work in the selected operating mode. The developed method takes into account losses and leaks in the engine starting system pipes. The data obtained using the developed method are source data for calculating and optimizing the air turbine's working process and for determining the time required to start the GTE, as well as for testing the operability of the GTE system by strength and other criteria. An algorithm for calculating the time required to start the GTE was also developed by the authors and implemented as a computer program. The obtained data can be used to analyze the possibility of starting the engine and to calculate its main parameters for specific elements of the engine starting system, to select the APP and SAT to meet the specifications.

The Interplay Between Loss Functions and Structural Constraints in Dependency Parsing

Dependency parsing can be cast as a combinatorial optimization problem with the objective to find the highest-scoring graph, where edge scores are learnt from data. Several of the decoding algorithms that have been applied to this task employ structural restrictions on candidate solutions, such as the restriction to projective dependency trees in syntactic parsing, or the restriction to noncrossing graphs in semantic parsing. In this paper we study the interplay between structural restrictions and a common loss function in neural dependency parsing, the structural hingeloss. We show how structural constraints can make networks trained under this loss function diverge and propose a modified loss function that solves this problem. Our experimental evaluation shows that the modified loss function can yield improved parsing accuracy, compared to the unmodified baseline.

Noncrossing partitions, noncrossing graphs, and q-permanental equations

We first prove a few simple results to illustrate some algebraic combinatorial features of the q-permanent. This is followed by a characterization of a noncrossing permutation in terms of the numbers of inversions of its cycles. Then we use a family of derivative formulas for the q-permanent of a square matrix A to characterize several structures of noncrossing kind. Each such formula f characterizes a set Df of digraphs, in the sense that D∈Df iff f is valid for all matrices A with digraph D. In this way we characterize, among others, digraphs with noncrossing permutation subdigraphs, noncrossing graphs, noncrossing forests. We use the derivative formulas to prove two particular cases of a conjecture on the q-monotonicity of the q-permanent of a Hermitian positive definite matrix.

Parsing to Noncrossing Dependency Graphs

We study the generalization of maximum spanning tree dependency parsing to maximum acyclic subgraphs. Because the underlying optimization problem is intractable even under an arc-factored model, we consider the restriction to noncrossing dependency graphs. Our main contribution is a cubic-time exact inference algorithm for this class. We extend this algorithm into a practical parser and evaluate its performance on four linguistic data sets used in semantic dependency parsing. We also explore a generalization of our parsing framework to dependency graphs with pagenumber at most k and show that the resulting optimization problem is NP-hard for k ≥ 2.

Maximal Rigid Objects as Noncrossing Bipartite Graphs

We classify the maximal rigid objects of the Σ2τ-orbit category \({\mathcal{C}}(Q)\) of the bounded derived category for the path algebra associated to a Dynkin quiver Q of type A, where τ denotes the Auslander-Reiten translation and Σ2 denotes the square of the shift functor, in terms of bipartite noncrossing graphs (with loops) in a circle. We describe the endomorphism algebras of the maximal rigid objects, and we prove that a certain class of these algebras are iterated tilted algebras of type A.

A characterization of horizontal visibility graphs and combinatorics on words

A Horizontal Visibility Graph (HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203–229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique, we determine asymptotically the average number of edges of HVGs.

Cyclic sieving for two families of non-crossing graphs

We prove the cyclic sieving phenomenon for non-crossing forests and non-crossing graphs. More precisely, the cyclic group acts on these graphs naturally by rotation and we show that the orbit structure of this action is encoded by certain polynomials. Our results confirm two conjectures of Alan Guo. Nous prouvons le phénomène de crible cyclique pour les forêts et les graphes sans croisement. Plus précisément, le groupe cyclique agit sur ces graphes naturellement par rotation et nous montrons que la structure d'orbite de cette action est codée par certains polynômes. Nos résultats confirment deux conjectures de Alan Guo.

Identities for non-crossing graphs and multigraphs

We refine an identity between the numbers of certain non-crossing graphs and multigraphs, by modifying a bijection found by P. Podbrdský [A bijective proof of an identity for noncrossing graphs, Discrete Math. 260 (2003) 249–253]. We also prove a new identity between the number of acyclic non-crossing graphs with n vertices and k edges (isolated vertices allowed and no multiple edges allowed), and the number of non-crossing connected graphs with n edges and k vertices (multiple edges allowed and no isolated vertices allowed).

On the Symmetry of the Distribution of $k$-Crossings and $k$-Nestings in Graphs

This note contains two results on the distribution of $k$-crossings and $k$-nestings in graphs. On the positive side, we exhibit a class of graphs for which there are as many $k$-noncrossing $2$-nonnesting graphs as $k$-nonnesting $2$-noncrossing graphs. This class consists of the graphs on $[n]$ where each vertex $x$ is joined to at most one vertex $y$ with $y < x$. On the negative side, we show that this is not the case if we consider arbitrary graphs. The counterexample is given in terms of fillings of Ferrers diagrams and solves a problem of Krattenthaler.

Noncrossing Trees and Noncrossing Graphs

We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of Panholzer and Prodinger for noncrossing trees and find a correspondence between a class of noncrossing trees, called proper noncrossing trees, and the set of symmetric ternary trees. The second result of this paper is a parity reversing involution on connected noncrossing graphs which leads to a relation between the number of noncrossing trees with $n$ edges and $k$ descents and the number of connected noncrossing graphs with $n+1$ vertices and $m$ edges.

The Non-Crossing Graph

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$

The Polytope of Non-Crossing Graphs on a Planar Point Set

For any set A of n points in ℝ2, we define a (3n - 3)-dimensional simple polyhedron whose face poset is isomorphic to the poset of ”non-crossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n - 3 where ni is the number of points of A in the interior of conv (A). The vertices of this polytope are all the pseudo-triangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudo-triangulations) and the removal or insertion of a single edge. As a by-product of our construction we prove that all pseudo-triangulations are infinitesimally rigid graphs.

Descents in Noncrossing Trees

The generating function for descents in noncrossing trees is found. A bijection shows combinatorially why the descent generating function with descents set equal to $2$ is the generating function for connected noncrossing graphs.

A bijective proof of an identity for noncrossing graphs

We give a bijective proof for the identity a n+2 =8 b n , where a n is the number of noncrossing simple graphs with n (possibly isolated) vertices and b n is the number of noncrossing graphs without isolated vertices and with n (possibly multiple) edges.