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Chessboard Complex
A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology. Informally, the (''m'', ''n'')-chessboard complex contains all sets of positions on an ''m''-by-''n'' chessboard, where Rook (chess), rooks can be placed without attacking each other. Equivalently, it is the matching complex of the (''m'', ''n'')-complete bipartite graph, or the independence complex of the ''m''-by-''n'' rook's graph. Definitions For any two positive integers ''m'' and ''n'', the (''m, n'')-chessboard complex \Delta_ is the abstract simplicial complex with vertex set [m]\times [n] that contains all subsets ''S'' such that, if (i_1,j_1) and (i_2,j_2) are two distinct elements of ''S'', then both i_1\neq i_2 and j_1\neq j_2. The vertex set can be viewed as a two-dimensional grid (a "chessboard"), and the complex contains all subsets ''S'' that do ''not'' contain two cells in the same row or in the same column. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |