Edge-isoperimetric problem: различия между версиями

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'''Edge-isoperimetric problem''' --- рёберно-изопериметрическая задача.  
'''Edge-isoperimetric problem''' рёберно-изопериметрическая задача.  


Given a graph <math>G = (V,E,\partial)</math> having the vertex-set <math>V</math>, edge-set <math>E</math> and boundary-function <math>\partial: \, E \rightarrow
Given a [[graph]] <math>G = (V,E,\partial)</math> having the vertex-set <math>V</math>, edge-set <math>E</math> and boundary-function <math>\partial: \, E \rightarrow
\left(\begin{array}{c} V \\ 2 \end{array}\right)</math> which identifies the
\left(\begin{array}{c} V \\ 2 \end{array}\right)</math> which identifies the
pair of vertices incident to each edge, we define
pair of [[vertex|vertices]] [[vertex incident to an edge|incident]] to each [[edge]], we define


<math>\Theta(S) = \{e \in E: \, \partial(e) = \{v,w\}, v \in S \& w \not \in S\}.</math>
<math>\Theta(S) = \{e \in E: \, \partial(e) = \{v,w\}, v \in S \And w \not \in S\}.</math>


Then, the '''edge-isoperimetric problem''' is to
Then, the '''edge-isoperimetric problem''' is to
minimize <math>|\Theta(S)|</math> over all <math>S \subseteq V</math> such that <math>|S| = k</math>  
minimize <math>|\Theta(S)|</math> over all <math>S \subseteq V</math> such that <math>|S| = k</math>  
for a given <math>k \in Z^{+}</math>.
for a given <math>k \in Z^{+}</math>.

Текущая версия от 11:33, 13 апреля 2011

Edge-isoperimetric problem — рёберно-изопериметрическая задача.

Given a graph [math]\displaystyle{ G = (V,E,\partial) }[/math] having the vertex-set [math]\displaystyle{ V }[/math], edge-set [math]\displaystyle{ E }[/math] and boundary-function [math]\displaystyle{ \partial: \, E \rightarrow \left(\begin{array}{c} V \\ 2 \end{array}\right) }[/math] which identifies the pair of vertices incident to each edge, we define

[math]\displaystyle{ \Theta(S) = \{e \in E: \, \partial(e) = \{v,w\}, v \in S \And w \not \in S\}. }[/math]

Then, the edge-isoperimetric problem is to minimize [math]\displaystyle{ |\Theta(S)| }[/math] over all [math]\displaystyle{ S \subseteq V }[/math] such that [math]\displaystyle{ |S| = k }[/math] for a given [math]\displaystyle{ k \in Z^{+} }[/math].