Comparability graph: различия между версиями
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'''Comparability graph''' | '''Comparability graph''' — ''[[граф сравнимости]].'' | ||
Let <math>G = (V,E)</math> be an undirected graph and let <math>F</math> be an | Let <math>\,G = (V,E)</math> be an [[undirected graph]] and let <math>\,F</math> be an ''orientation'' of its [[edge|edges]] (i.e. <math>\,(V,F)</math> is the resulting [[oriented graph]]). | ||
''orientation'' of its edges (i.e. <math>(V,F)</math> is the resulting oriented graph). | <math>\,F</math> is called a '''[[transitive orientation]]''' of <math>\,G</math> if the following | ||
<math>F</math> is called a '''transitive orientation''' of <math>G</math> if the following | |||
properties hold: | properties hold: | ||
<math>F \cap F^{-1} = \emptyset\mbox{ and }F + F^{-1} = E\mbox{ and }F^{2} \subseteq F,</math> | <math>\,F \cap F^{-1} = \emptyset\mbox{ and }F + F^{-1} = E\mbox{ and }F^{2} \subseteq F,</math> | ||
where <math>F^{2} = \{(a,c)| \; \exists_{b \in V}(a,b) \in F \wedge (b,c) | where <math>\,F^{2} = \{(a,c)| \; \exists_{b \in V}(a,b) \in F \wedge (b,c) \in F\}</math>. | ||
\in F\}</math>. | |||
A graph <math>G</math> which admits a '''transitive orientation''' of its edges is called | A [[graph, undirected graph, nonoriented graph|graph]] <math>\,G</math> which admits a '''transitive orientation''' of its edges is called a '''comparability graph'''. | ||
a '''comparability graph'''. | |||
If graph <math>G</math> is a '''comparability''' graph, then this also holds for every induced | If graph <math>\,G</math> is a '''comparability''' graph, then this also holds for every [[induced | ||
subgraph of <math>G</math>. | subgraph]] of <math>\,G</math>. | ||
The other name is a '''Transitively orientable graph'''. | The other name is a '''[[Transitively orientable graph]]'''. | ||
==Литература== | |||
* Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009. | |||
Текущая версия от 10:46, 24 октября 2018
Comparability graph — граф сравнимости.
Let [math]\displaystyle{ \,G = (V,E) }[/math] be an undirected graph and let [math]\displaystyle{ \,F }[/math] be an orientation of its edges (i.e. [math]\displaystyle{ \,(V,F) }[/math] is the resulting oriented graph). [math]\displaystyle{ \,F }[/math] is called a transitive orientation of [math]\displaystyle{ \,G }[/math] if the following properties hold:
[math]\displaystyle{ \,F \cap F^{-1} = \emptyset\mbox{ and }F + F^{-1} = E\mbox{ and }F^{2} \subseteq F, }[/math]
where [math]\displaystyle{ \,F^{2} = \{(a,c)| \; \exists_{b \in V}(a,b) \in F \wedge (b,c) \in F\} }[/math].
A graph [math]\displaystyle{ \,G }[/math] which admits a transitive orientation of its edges is called a comparability graph.
If graph [math]\displaystyle{ \,G }[/math] is a comparability graph, then this also holds for every [[induced subgraph]] of [math]\displaystyle{ \,G }[/math].
The other name is a Transitively orientable graph.
Литература
- Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.