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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Graphic matroid ： ウィキペディア英語版 Graphic matroid In matroid theory, a discipline within mathematics, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids.〔 uses a reversed terminology, in which he called bond matroids "graphic" and cycle matroids "co-graphic", but this has not been followed by later authors.〕 A matroid that is both graphic and co-graphic is called a planar matroid; these are exactly the graphic matroids formed from planar graphs. ==Definition== A matroid may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets $A$ and $B$ are both independent, and $A$ is larger than $B$, then there is an element $x\backslash in\; A\backslash setminus\; B$ such that $B\backslash cup\backslash$ remains independent. If $G$ is an undirected graph, and $F$ is the family of sets of edges that form forests in $G$, then $F$ is clearly closed under subsets (removing edges from a forest leaves another forest). It also satisfies the exchange property: if $A$ and $B$ are both forests, and $A$ has more edges than $B$, then it has fewer connected components, so by the pigeonhole principle there is a component $C$ of $A$ that contains vertices from two or more components of $B$. Along any path in $C$ from a vertex in one component of $B$ to a vertex of another component, there must be an edge with endpoints in two components, and this edge may be added to $B$ to produce a forest with more edges. Thus, $F$ forms the independent sets of a matroid, called the graphic matroid of $G$ or $M(G)$. More generally, a matroid is called graphic whenever it is isomorphic to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph.〔 The bases of a graphic matroid $M(G)$ are the spanning forests of $G$, and the cycles of $M(G)$ are the simple cycles of $G$. The rank in $M(G)$ of a set $X$ of edges of a graph $G$ is $r(X)=n-c$ where $n$ is the number of vertices in the subgraph formed by the edges in $X$ and $c$ is the number of connected components of the same subgraph.〔 The corank of the graphic matroid is known as the circuit rank or cyclomatic number. 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「Graphic matroid」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.