Asteroidal number: различия между версиями
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'''Asteroidal number''' — ''[[астероидальное число]].'' | '''Asteroidal number''' — ''[[астероидальное число]].'' | ||
A set of [[vertex|vertices]] <math>A \subseteq V</math> of a [[graph, undirected graph, nonoriented graph|graph]] <math>G = (V,E)</math> is an '''asteroidal set''' if for each <math>a \in A</math> the set <math>A - a</math> is contained in one | A set of [[vertex|vertices]] <math>A \subseteq V</math> of a [[graph, undirected graph, nonoriented graph|graph]] <math>\,G = (V,E)</math> is an '''asteroidal set''' if for each <math>a \in A</math> the set <math>\,A - a</math> is contained in one | ||
component of <math>G - N[a]</math>. The '''asteroidal number''' of a graph <math>G</math>, denoted by | component of <math>\,G - N[a]</math>. The '''asteroidal number''' of a graph <math>\,G</math>, denoted by | ||
<math>an(G)</math>, is the maximum cardinality of the asteroidal set in <math>G</math>. | <math>\,an(G)</math>, is the maximum cardinality of the asteroidal set in <math>\,G</math>. | ||
Graphs with '''asteroidal number''' at most two are commonly known as '''AT-free graphs'''. The | Graphs with '''asteroidal number''' at most two are commonly known as '''AT-free graphs'''. The | ||
class of AT-free graphs contains well-known graph classes such as ''[[interval graph|interval]], [[permutation graph|permutation]]'' and ''[[cocomparability graph|cocomparability]]'' graphs. | class of AT-free graphs contains well-known graph classes such as ''[[interval graph|interval]], [[permutation graph|permutation]]'' and ''[[cocomparability graph|cocomparability]]'' graphs. | ||
Версия от 11:48, 9 декабря 2011
Asteroidal number — астероидальное число.
A set of vertices [math]\displaystyle{ A \subseteq V }[/math] of a graph [math]\displaystyle{ \,G = (V,E) }[/math] is an asteroidal set if for each [math]\displaystyle{ a \in A }[/math] the set [math]\displaystyle{ \,A - a }[/math] is contained in one component of [math]\displaystyle{ \,G - N[a] }[/math]. The asteroidal number of a graph [math]\displaystyle{ \,G }[/math], denoted by [math]\displaystyle{ \,an(G) }[/math], is the maximum cardinality of the asteroidal set in [math]\displaystyle{ \,G }[/math].
Graphs with asteroidal number at most two are commonly known as AT-free graphs. The class of AT-free graphs contains well-known graph classes such as interval, permutation and cocomparability graphs.