Michal Adamaszek
We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.
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A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented. In addition, it will be shown that a quasi-forest is shellable if and only if its $h$-vector $(h_0, h_1, h_2, ...)$ satisfies $h_i = 0$ for $i > 1$.
Skeleton Chordalities
研究图弦性新的高维类似物，给出单纯复形相关主要结果，如骨架 - E - 弦性、顶点可分解性、团弦性等之间的蕴含关系及等价关系，还包括复形分解、单纯顶点等性质，扩展了相关工作。
We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.
Let $G$ be a graph on $V$. A vertex subset $S \subset V$ is called a cover of $G$ if its complement is an independent set, and $S$ is called a noncover if it is not a cover of $G$. A noncover complex $NC(G)$ of $G$ is the simplicial complex on $V$ whose faces are noncovers of $G$. The independence domination number $ iγ(G)$ of $G$ is the minimum integer $k$ such that every independent set of $G$ can be dominated by $k$ vertices. In this note, we prove that $NC(G)$ is $(|V|- iγ(G)-1)$-collapsible.
A recent conjecture that appeared in three papers by Bigdeli--Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the $k$-skeleton of the simplex is extendably shellable, for any $k$. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.
Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $β(I(G)) = (β_0, β_1, ..., β_p)$ denote the Betti sequence of $I(G)$, where $β_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the projective dimension of $I(G)$. It will be shown that there exists a simplicial complex $Δ$ of dimension $p$ whose $f$-vector $f (Δ) = (f_0, f_1, ..., f_p)$ coincides with $β(I(G))$.
Vertex decomposable complexes of directed forests, conflict graphs and chordality
本文研究了有向森林的顶点可分解复形，通过构造冲突图并应用独立复形理论，得出了保证复形顶点可分解、壳性和序列Cohen-Macaulay性质的图论准则，并给出了相关多重有向图的分类。
On the Vertex Decomposability of $r$-Independence Complexes of Trees
本文证明了对于任意树 $T$ 和整数 $r \ge 1$，$r$-独立复形 $\Ind_r(T)$ 是顶点可分解的，从而解决了 Abdelmalek 等人的一个猜想。
In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. In \cite{F1}, Fröberg shows that the chordal graphs corresponds to graph algebras, $R/I(\mc{G})$, with linear resolutions. We extend Fröberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The defini…
By using Alexander duality on simplicial complexes we give a new and algebraic proof of Dirac's theorem on chordal graphs.
Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs
本文证明共区间超图与下闭复形在补集意义下等价，并证明下闭团簇是Woodroofe弦性的，从而回答Dochtermann和Engström的问题，表明其Alexander对偶复形是顶点可分解的，且具有由下闭顶点序诱导的壳化。
Chordal graphs, higher independence and vertex decomposable complexes
该研究推广独立复形提出$r$-独立复形，证明树的$r$-独立复形可剥壳、毛虫图的$r$-独立复形顶点可分解，还构造出$r$-独立复形非序列Cohen-Macaulay的弦图，推动了弦图与拓扑复形性质的交叉研究。
We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and they are a rich new class of simplicial complexes. We show that Dirac's theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley's reformulation of Dirac's theorem on chordal graphs.
Helly complexes are Hellyfications of their boundaries
本文证明有限 Helly 复形的一维骨架同构于其边界的 Hellyfication，为边界刚性现象提供典范内射包解释。
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
A graph $G$ is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call `edge-erasures'. We show that these moves are in fact equivalent to a linear quotient ordering on $I_{\overline{G}}$, the edge ideal of the complement graph. Known results imply that $I_{\overline G}$ has linear quotients if and only if $G$ is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higher-dimensional analogues of thi…
For $r\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this article, we establish a relation between the distance $r$-domination number of $G$ and (homological) connectivity of Ind$_r(G)$. We also prove that Ind$_r(G)$, for a chordal graph $G$, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose $r$-independence complex ha…
Stanley-Reisner ideals of higher independence complexes of chordal graphs
本文研究 chordal 图的 t-独立复形的 Stanley-Reisner 理想，证明其正则度和射影维数的公式，并探讨其线性解和 Cohen-Macaulay 性质的条件。
Given a graph $G$, the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i γ(G)-1)$-collapsible where $i γ(G)$ denotes the independent domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs. Namely, we show that for a graph $G$, the non-cover complex of a graph $G$ is $(|V(G)|-i γ(G)-1)$-collapsible.
Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals
研究Chordal Clique Complexes的同调性质，通过计算f向量和Hilbert级数，得到所有k-骨架的显式分层Betti数，并揭示其正则度和射影维数的性质，同时推导出对偶体的f向量、有理h多项式和分层Betti数。
Double cohomology of moment-angle complexes
本文在矩角复形的上同调上定义双上同调结构，给出微分的三个等价定义，并计算包含弦图团复形的单纯复形族的双上同调。
We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients of quadratic monomial ideals.
The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.
Shellability in Clique-Free Complexes of Graphs
本文研究了$t$-无团复形的壳性，证明了特定条件下的复形可分解且壳性，并引入了团附加操作以构造壳性复形，同时扩展了团尾巴的概念并建立了弦性与线性解的关系。
Generalizations of interval and proper interval graphs for simplicial complexes
本文将区间图和真区间图推广到单纯复形，给出组合与代数刻画，并应用于可排序性、Cohen-Macaulay 整环及禁子图性质。
Total Cut Complexes of Graphs
本文受Fröberg及Eagon和Reiner工作启发，定义图的全k-割复形，并利用代数拓扑和离散Morse理论研究弦图、圈、二分图、棱柱图及网格图等图类的同伦类型和组合性质。
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…
The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem
本文研究了 r-连通子图复形的拓扑与代数性质，揭示了其与图类 co-chordality 的关系，并扩展了 Fröberg 定理。
CAT(0) square complexes that do not embed into finite products of trees
针对Chepoi和Hagen的问题，构造了两个有界度CAT(0)方形复形，证明其不能等距嵌入树的有限乘积，并揭示Burling图是唯一障碍，同时构造了高围长高色数的弦图与区间图交集。
Completing and extending shellings of vertex decomposable complexes
本文证明顶点可分解复形满足 Simon 猜想中的剥壳可补全性质，并给出通过按反字典序添加缺失子集保持顶点可分解性的构造方法，同时探讨其在拟阵、移位复形及小顶点数情形下的应用与等价关系。
A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex
本文提出一种递归方法，用于构建图的独立复形上的无环匹配，应用离散Morse理论确定了几种图类的同调类型，并为高效同调计算提供预处理框架。
Fröberg's Theorem, vertex splittability and higher independence complexes
该研究将Fröberg定理推广至高次无平方单项式理想，通过r-独立构造r-均匀超图，证明共弦图对应边理想具顶点可分裂性，计算了分次Betti数，并给出线性消解存在性的拓扑证明。
Vertex cut of a graph and connectivity of its neighbourhood complex
本文研究图的邻域复形连通性与顶点连通性的关系，提出顶点扩张条件下的连通性保持定理，并应用于弦图、弱三角化图及 queen、king 图。
A Complete Classification of 2-Linear Neighborhood Complexes
本文刻画了图G的Stanley-Reisner环k[N(G)]何时具有2-线性分解，等价于G是邻域共形且其公共邻域图是弦图，即G是仅含4-圈作为诱导圈的二分图，并由此得到Katzman下界等号成立条件及Cohen-Macaulay支配复形的分类。
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