A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled speed of P.
The possible labelled speeds of a hereditary property of graphs have been extensively studied, and the aim of this paper is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labelled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2^n - 1. We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of posets, tournaments and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and M…
A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield to quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $…
Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support $τ$-tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support $τ$-tilting posets coincide with those of tree quiver algebras and give a full…
We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. By relying on the interplay between bipartite distance hereditary graphs and series-parallel graphs, we show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal tim…
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.
Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers -1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.
If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the …
The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors, evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, we prove that the one-variable vertex-nullity interlace polynomial is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tut…
Several graph properties are characterized as the class of graphs that admit an orientation avoiding finitely many oriented structures. For instance, if $F_k$ is the set of homomorphic images of the directed path on $k+1$ vertices, then a graph is $k$-colourable if and only if it admits an orientation with no induced oriented graph in $F_k$. There is a fundamental question underlying this kind of characterizations: given a graph property, $\mathcal{P}$, is there a finite set of oriented graphs, $F$, such that a graph belongs to $\mathcal{P}$ if and only if it admits an orientation with no indu…
Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary $\calL$-properties, where $\calL$ is any finite relational language with maximum arity at le…
In contrast to the abundance of "direct" Ramsey results for classes of finite structures (such as finite ordered graphs, finite ordered metric spaces and finite posets with a linear extension), in only a handful of cases we have a meaningful dual Ramsey result. In this paper we prove a dual Ramsey theorem for finite ordered oriented graphs. Instead of embeddings, which are crucial for "direct" Ramsey results, we consider a special class of surjective homomorphisms between finite ordered oriented graphs. Since the setting we are interested in involves both structures and morphisms, all our resu…
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids…
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…
In recent breakthrough results, Saxton--Thomason and Balogh--Morris--Samotij have developed powerful theories of hypergraph containers. These theories have led to a large number of new results on transference, and on counting and characterising typical graphs in hereditary properties. In a different direction, Hatami--Janson--Szegedy proved results on the entropy of graph limits which count and characterise graphs in dense hereditary properties. In this paper, we make a threefold contribution to this area of research: 1) We generalise results of Saxton--Thomason to obtain container theorems fo…
We study the directed maximum common edge subgraph problem (DMCES) for the class of directed graphs that are finite, weakly connected, oriented, and simple. We use DMCES to define a metric on partially ordered sets that can be represented as weakly connected directed acyclic graphs. While most existing metrics assume that the underlying sets of the partial order are identical, and only the relationships between elements can differ, the metric defined here allows the partially ordered sets to be different. The proof that there is a metric based on DMCES involves the extension of the concept of …
In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \in V $ a set $S_i \in Z$ such that $v_i v_j \in E$ if and only if $S_i \subset S_j$ or $S_j \subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \in V $ we can assign a set $S_i \in Z$ such that $v_i < v_j$ if and only if $S_i \subset S_j$. Obviously, …
The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Brüstle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graph…
In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a combinatorial model for torsion pairs using polyogonal subdivisions of a convex polygon. We use this model and the lattice theory to classify 2-term simple-minded collections in the bounded derived category of the corresponding tiling algebra. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver. Our m…
Monadic dependence from reducts, and applications to twin-width of oriented graphs
Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息学研究与综合实验室）；Centre d'Analyse et de Mathématique Sociales CNRS UMR 8557（社会分析与数学中心）；Southeast University（东南大学）；Computer Science Institute of Charles University (IUUK)（查理大学计算机科学研究所）
An interval in a combinatorial structure S is a set I of points which relate to every point from S I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes -- this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f(n) elements. We prove such results when C is the class of all t…
