Several Convex-Ear Decompositions

凸多边形的最小单纯形带符号Minkowski分解及max-plus函数的分解

Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions

本文研究了多胞体的带符号Minkowski分解与max-plus函数分解的对应关系，证明了2维整点多边形可分解为整点、长度1的整线段和面积1/2的整三角形，并据此得出二元整数系数max-plus函数可表示为更简单函数的集合。

拟阵的增广Bergman复形的凸耳分解

A convex ear decomposition of the augmented Bergman complex of a matroid

本文证明任意拟阵的增广Bergman复形具有凸耳分解，进而推出其双重Cohen–Macaulay性与顶部重h-向量，并给出h-多项式计算公式及非对数凹、非单峰的反例。

Minimal Convex Decompositions

Let $P$ be a set of $n$ points on the plane in general position. We say that a set $Γ$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $Γ$ is the convex hull of $P,$ every element in $Γ$ is empty, and for any two different elements of $Γ$ their interiors are disjoint. A minimal convex decomposition of $P$ is a convex decomposition $Γ'$ such that for any two adjacent elements in $Γ'$ its union is a non convex polygon. It is known that $P$ always has a minimal convex decomposition with at most $\frac{3n}{2}$ elements. Here we prove that $P$ a…

Integer decomposition property of free sums of convex polytopes

Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.

Decompositions of a polygon into centrally symmetric pieces

In this paper we deal with edge-to-edge, irreducible decompositions of a centrally symmetric convex $(2k)$-gon into centrally symmetric convex pieces. We prove an upper bound on the number of these decompositions for any value of $k$, and characterize them for octagons.

Ear Decomposition and Balanced Neighborly Simplicial Manifolds

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.

On Ear Decompositions of Strongly Connected Bidirected Graphs

Bidirected graphs (earlier studied by Edmonds, Johnson and, in equivalent terms of skew-symmetric graphs, by Tutte, Goldberg, Karzanov, and others) proved to be a useful unifying language for describing both flow and matching problems. In this paper we extend the notion of ear decomposition to the class of strongly connected bidirected graphs. In particular, our results imply Two Ear Theorem on matching covered graphs of Lovász and Plummer. The proofs given here are self-contained except for standard Barrier Theorem on skew-symmetric graphs.

Peripheral convex expansions of resonance graphs

In this paper, we show that the resonance graph of a plane elementary bipartite graph $G$ can be obtained from an edge by a sequence of peripheral convex expansions with respect to a reducible face decomposition of $G$ if and only if the infinite face of $G$ is forcing.

Homometry and direct-sum decompositions of lattice-convex sets

Two sets in $\mathbb{R}^d$ are called homometric if they have the same covariogram, where the covariogram of a finite subset $K$ of $\mathbb{R}^d$ is the function associating to each $u \in \mathbb{R}^d$ the cardinality of $K \cap (K+u)$. Understanding the structure of homometric sets is important for a number of areas of mathematics and applications. If two sets are homometric but do not coincide up to translations and point reflections, we call them nontrivially homometric. We study nontrivially homometric pairs of lattice-convex sets, where a set $K$ is called lattice-convex with respect to…

凸分解空间与 Crapo 补公式

Convex decomposition spaces and Crapo complementation formula

本文在一般分解空间中引入并发展凸性概念，建立了 Möbius 函数的 Crapo 补公式，并在 objective 层面给出 ∞-群胚同伦等价的显式证明，统一推广了 Björner–Walker 公式和有限格上的原始 Crapo 补公式。

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomi…

Decomposition spaces and restriction species

We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and…

Regularity of the vanishing ideal over a bipartite nested ear decomposition

We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ ε-3}{2}(q-2),$$ …

g-elements, finite buildings and higher Cohen-Macaulay connectivity

The main result is a proof that the g-vector of a simplicial complex with a convex ear decomposition is an M-vector. This is a generalization of similar results for matroid complexes. We also show that a finite building has a convex ear decomposition. This leads to connections between higher Cohen-Macaulay connectivity and increasing h-vectors.

无向图的高效递归分解算法

An efficient recursive decomposition algorithm for undirected graphs

本文提出了一种高效递归分解算法，用于无向图的原子分解，通过MCS顺序和凸扩展方法提升分解效率。

Associated primes of powers of edge ideals and ear decompositions of graphs

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.