Clique-transversal: различия между версиями
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'''Clique-transversal''' | '''Clique-transversal''' — ''[[кликовая трансверсаль]].'' | ||
A '''clique-transversal''' of a graph <math>G</math> is a subset of vertices that | A '''clique-transversal''' of a [[graph, undirected graph, nonoriented graph|graph]] <math>\,G</math> is a subset of [[vertex|vertices]] that meets all the [[clique|cliques]]. A '''clique-independent set''' is a collection of pairwise vertex disjoint cliques. The '''[[clique-transversal number]]''' and '''[[clique-independence number]]''' of <math>\,G</math>, denoted by <math>\,\tau_{c}(G)</math> and <math>\,\alpha_{c}(G)</math>, are the sizes of a | ||
meets all the cliques. A '''clique-independent set''' is a collection | minimum clique-transversal and a maximum [[clique-independent set]] of <math>\,G</math>, | ||
of pairwise vertex disjoint cliques. The '''clique-transversal number''' and '''clique-independence number''' of <math>G</math>, denoted by | |||
<math>\tau_{c}(G)</math> and <math>\alpha_{c}(G)</math>, are the sizes of a | |||
minimum clique-transversal and a maximum clique-independent set of <math>G</math>, | |||
respectively. | respectively. | ||
It is easy to see that <math>\tau_{c}(G) \geq \alpha_{c}(G)</math> for any graph | It is easy to see that <math>\tau_{c}(G) \geq \alpha_{c}(G)</math> for any graph <math>\,G</math>. A graph <math>\,G</math> is '''[[clique-perfect graph|clique-perfect]]''' if <math>\,\tau_{c}(H) = | ||
<math>G</math>. A graph <math>G</math> is '''clique-perfect''' if <math>\tau_{c}(H) = | \alpha_{c}(H)</math> for every induced [[subgraph]] <math>\,H</math> of <math>\,G</math>. If this equality holds for the graph <math>\,G</math>, we say that <math>\,G</math> is '''[[clique-good graph|clique-good]]'''. | ||
\alpha_{c}(H)</math> for every induced subgraph <math>H</math> of <math>G</math>. If this equality | |||
holds for the graph <math>G</math>, we say that <math>G</math> is '''clique-good'''. | ==Литература== | ||
* Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009. | |||
Текущая версия от 10:44, 24 октября 2018
Clique-transversal — кликовая трансверсаль.
A clique-transversal of a graph [math]\displaystyle{ \,G }[/math] is a subset of vertices that meets all the cliques. A clique-independent set is a collection of pairwise vertex disjoint cliques. The clique-transversal number and clique-independence number of [math]\displaystyle{ \,G }[/math], denoted by [math]\displaystyle{ \,\tau_{c}(G) }[/math] and [math]\displaystyle{ \,\alpha_{c}(G) }[/math], are the sizes of a minimum clique-transversal and a maximum clique-independent set of [math]\displaystyle{ \,G }[/math], respectively.
It is easy to see that [math]\displaystyle{ \tau_{c}(G) \geq \alpha_{c}(G) }[/math] for any graph [math]\displaystyle{ \,G }[/math]. A graph [math]\displaystyle{ \,G }[/math] is clique-perfect if [math]\displaystyle{ \,\tau_{c}(H) = \alpha_{c}(H) }[/math] for every induced subgraph [math]\displaystyle{ \,H }[/math] of [math]\displaystyle{ \,G }[/math]. If this equality holds for the graph [math]\displaystyle{ \,G }[/math], we say that [math]\displaystyle{ \,G }[/math] is clique-good.
Литература
- Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.