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:''“Whitney complex” redirects here. For the Mississippi sports facility, see Davey Whitney Complex.'' Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. The clique complex ''X''(''G'') of an undirected graph ''G'' is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of ''G''. Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of ''k'' vertices is represented by a simplex of dimension ''k'' − 1. The 1-skeleton of ''X''(''G'') (also known as the ''underlying graph'' of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2-element set in the family; it is isomorphic to ''G''.〔.〕 Clique complexes are also known as Whitney complexes. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph ''G'' onto the manifold in such a way that every face is a triangle and every triangle is a face. If a graph ''G'' has a Whitney triangulation, it must form a cell complex that is isomorphic to the Whitney complex of ''G''. In this case, the complex (viewed as a topological space) is homeomorphic to the underlying manifold. A graph ''G'' has a 2-manifold clique complex, and can be embedded as a Whitney triangulation, if and only if ''G'' is locally cyclic; this means that, for every vertex ''v'' in the graph, the induced subgraph formed by the neighbors of ''v'' forms a single cycle.〔; ; .〕 ==Independence complex== The independence complex ''I''(''G'') of a graph ''G'' is formed in the same way as the clique complex from the independent sets of ''G''. It is the clique complex of the complement graph of ''G''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Clique complex」の詳細全文を読む スポンサード リンク
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