一直以来，纯shellable性质因其作为通往Cohen-Macaulay性质的一条重要途径而吸引着众多组合和代数学家的研究兴趣。本项目申请人及其合作者在前期工作中通过增加约束条件的方法引入了强shellable复形的定义，并围绕强shellable性质展开了系列研究。本项目拟在此基础上，借助余一维图这一新引进的图论工具，结合单纯复形的组合属性与其Stanley-Reisner理想的代数属性之间的对应关系，来探讨强shellable复形、bi-SS复形（其本身与其Alexander对偶都满足强shellable性质的复形）的组合性质，以及其toric理想的代数性质。通过本项目的研究，我们希望结合图论的直观性、组合的技巧性与代数的深刻性，将强shellable复形、bi-SS复形的潜在内涵挖掘出来，进一步丰富和完善与shellable性质、Cohen-Macaulay性质相关的理论体系。

As we know, shellability is an important property which attracts so many combinatorial algebraists, since a pure shellable complex is Cohen-Macaulay. By adding a new restriction condition to the definition of shellability, we introduced a special shellable complex, called a strongly shellable complex. Based on a series of previous research about strong shellability, in this research project, we plan to study the combinatorial and algebraic properties of strongly shellable complexes and bi-SS complexes, where a bi-SS complex is a simplicial complex which as well as its Alexander dual are strongly shellable. In this process, we are making use of two important tools. One is the codimension one graph of a strongly shellable complex, while the other is about the relation between the combinatorial properties of a simplicial complex and the algebraic properties of the corresponding Stanley-Reisner ideal..By the work of this research project, we are making effort to revealing the internal properties of the objects related to strong shellability, which will enrich the theory of combinatorial commutative algebra and promote the permeation and integration of algebra, combinatorics and graph theory.