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rigidity matroid
In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with ''n'' vertices in ''d''-dimensional space, a set of edges that defines a subgraph with ''k'' degrees of freedom has matroid rank ''dn'' − ''k''. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.〔.〕〔.〕〔.〕
==Definition==
A ''framework'' is an undirected graph, embedded into ''d''-dimensional Euclidean space by providing a ''d''-tuple of Cartesian coordinates for each vertex of the graph. From a framework with ''n'' vertices and ''m'' edges, one can define a matrix with ''m'' rows and ''nd'' columns, an expanded version of the incidence matrix of the graph called the ''rigidity matrix''. In this matrix, the entry in row ''e'' and column (''v'',''i'') is zero if ''v'' is not an endpoint of edge ''e''. If, on the other hand, edge ''e'' has vertices ''u'' and ''v'' as endpoints, then the value of the entry is the difference between the ''i''th coordinates of ''v'' and ''u''.〔〔
The rigidity matroid of the given framework is a linear matroid that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is linearly independent. A framework is called ''generic'' if the coordinates of its vertices are algebraically independent real numbers. Any two generic frameworks on the same graph ''G'' determine the same rigidity matroid, regardless of their specific coordinates. This is the (''d''-dimensional) rigidity matroid of ''G''.〔〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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