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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 bidimensionality ： ウィキペディア英語版 bidimensionality Bidimensionality theory characterizes a broad range of graph problems (bidimensional) that admit efficient approximate, fixed-parameter or kernel solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the graph minor theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. The theory was introduced in the work of Demaine, Fomin, Hajiaghayi, and Thilikos, for which the authors received the Nerode Prize in 2015. ==Definition== A parameterized problem $\backslash Pi$ is a subset of $\backslash Gamma^\backslash times\; \backslash mathbb$ for some finite alphabet $\backslash Gamma$. An instance of a parameterized problem consists of ''(x,k)'', where ''k'' is called the parameter. A parameterized problem $\backslash Pi$ is ''minor-bidimensional'' if # For any pair of graphs $H,G$, such that $H$ is a minor of $G$ and integer $k$, $(G,k)\backslash in\; \backslash Pi$ yields that $(H,k)\backslash in\; \backslash Pi$. In other words, contracting or deleting an edge in a graph $G$ cannot increase the parameter; and # there is $\backslash delta\; >\; 0$ such that for every $(r\; \backslash times\; r)$-grid $R$, $(R,\; k)\backslash not\; \backslash in\; \backslash Pi$ for every $k\backslash leq\; \backslash delta\; r^2$. In other words, the value of the solution on $R$ should be at least $\backslash delta\; r^2$. Examples of minor-bidimensional problems are the parameterized versions of vertex cover, feedback vertex set, minimum maximal matching, and longest path. Let $\backslash Gamma\_$ be the graph obtained from the $(r\backslash times\; r)$-grid by triangulating internal faces such that all internal vertices become of degree 6, and then one corner of degree two joined by edges with all vertices of the external face. A parameterized problem $\backslash Pi$ is ''contraction-bidimensional'' if # For any pair of graphs $H,G$, such that $H$ is a contraction of $G$ and integer $k$, $(G,k)\backslash in\; \backslash Pi$ yields that $(H,k)\backslash in\; \backslash Pi$. In other words, contracting an edge in a graph $G$ cannot increase the parameter; and # there is $\backslash delta\; >\; 0$ such that $(\backslash Gamma\_r,\; k)\backslash not\; \backslash in\; \backslash Pi$ for every $k\backslash leq\; \backslash delta\; r^2$. Examples of contraction-bidimensional problems are dominating set, connected dominating set, max-leaf spanning tree, and edge dominating set. 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「bidimensionality」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.