Bisection width of a graph: различия между версиями

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'''Bisection width of a graph''' --- ширина бисекции графа.  
'''Bisection width of a graph''' — ''[[ширина бисекции графа]].''


The '''bisection width''' <math>bw(G)</math> '''of a graph''' <math>G</math> is the minimal number of edges
The '''bisection width''' <math>\,bw(G)</math> '''of a graph''' <math>\,G</math> is the minimal number of [[edge|edges]]
between vertex sets <math>A</math> and <math>\bar{A}</math> of almost equal sizes, i.e. <math>A
between [[vertex]] sets <math>\,A</math> and <math>\bar{A}</math> of almost equal sizes, i.e. <math>A
\cup \bar{A} = V(G)</math> and <math>||A| - |\bar{A}|| \leq 1</math>. If <math>A \subseteq
\cup \bar{A} = V(G)</math> and <math>||A| - |\bar{A}|| \leq 1</math>. If <math>A \subseteq
V(G)</math>, then <math>E(A,\bar{A})</math> denotes the set of edges of <math>G</math> having one
V(G)</math>, then <math>E(A,\bar{A})</math> denotes the set of edges of <math>\,G</math> having one
end in <math>A</math> and another end in <math>V(G) \setminus A = \bar{A}</math>. The '''isoperimetric number''' <math>i(G)</math> of a graph <math>G</math> equals the minimum
end in <math>\,A</math> and another end in <math>V(G) \setminus A = \bar{A}</math>. The '''[[isoperimetric number]]''' <math>\,i(G)</math> of a [[graph, undirected graph, nonoriented graph|graph]] <math>\,G</math> equals the minimum
of the ratio <math>|E(A,\bar{A}|/|A|</math> for all <math>A \subseteq V(G)</math> such that
of the ratio <math>|E(A,\bar{A}|/|A|</math> for all <math>A \subseteq V(G)</math> such that
<math>2|A| \leq |V(G)| = n</math>.
<math>2|A| \leq |V(G)| = n</math>.
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characteristics:
characteristics:


<math>i(G) \leq \frac{2}{n}bw(G).</math>
:::::<math>i(G) \leq \frac{2}{n}bw(G).</math>
 
==Литература==
 
* Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.

Текущая версия от 16:31, 23 октября 2018

Bisection width of a graphширина бисекции графа.

The bisection width [math]\displaystyle{ \,bw(G) }[/math] of a graph [math]\displaystyle{ \,G }[/math] is the minimal number of edges between vertex sets [math]\displaystyle{ \,A }[/math] and [math]\displaystyle{ \bar{A} }[/math] of almost equal sizes, i.e. [math]\displaystyle{ A \cup \bar{A} = V(G) }[/math] and [math]\displaystyle{ ||A| - |\bar{A}|| \leq 1 }[/math]. If [math]\displaystyle{ A \subseteq V(G) }[/math], then [math]\displaystyle{ E(A,\bar{A}) }[/math] denotes the set of edges of [math]\displaystyle{ \,G }[/math] having one end in [math]\displaystyle{ \,A }[/math] and another end in [math]\displaystyle{ V(G) \setminus A = \bar{A} }[/math]. The isoperimetric number [math]\displaystyle{ \,i(G) }[/math] of a graph [math]\displaystyle{ \,G }[/math] equals the minimum of the ratio [math]\displaystyle{ |E(A,\bar{A}|/|A| }[/math] for all [math]\displaystyle{ A \subseteq V(G) }[/math] such that [math]\displaystyle{ 2|A| \leq |V(G)| = n }[/math].

From the definitions, we have the following inequality for these characteristics:

[math]\displaystyle{ i(G) \leq \frac{2}{n}bw(G). }[/math]

Литература

  • Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.