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Difference between revisions of "Threshold graph"

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(Start article: Threshold graph)
(characterisations, cite Golumbic & Trenk (2004))
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{{MSC|05C}}
 
{{MSC|05C}}
  
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A graph with [[Dilworth number]] $1$: for any two vertices $x,y$, one of the neighbourhoods $N(x)$, $N(y)$ contains the other.  Such graphs are characterised by having no induced subgraph of the form $K_{2,2}$ (complete bipartite on $2+2$ points) , $C_4$ (cycle of length $4$) or $P_4$ (path of length $4$). They are the [[comparability graph]]s of [[threshold order]]s.
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A finite unoriented graph $G=(V,E)$ with a weight function $w : V \rightarrow \mathbf{R}$ and a threshold value $T$ such that a set $S$ of vertices is independent (has no edges) if and only if $\sum_{v \in S} w(v) < T$.
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The following properties characterise threshold graphs:
  
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There is a polynomial time algorithm for computing the Dilworth number of a finite graph.
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* They have [[Dilworth number]] $1$: for any two vertices $x,y$, one of the neighbourhoods $N(x)$, $N(y)$ contains the other. 
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* They have no induced subgraph of the form $K_{2,2}$ (complete bipartite on $2+2$ points) , $C_4$ (cycle of length $4$) or $P_4$ (path of length $4$). 
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* They are the [[comparability graph]]s of [[threshold order]]s.
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A graph $G$ is a threshold graph if and only if the [[graph complement]] $\bar G$ is a threshold graph.
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There is a polynomial time algorithm for computing the Dilworth number of a finite graph and so for recognising a threshold graph.
  
 
==References==
 
==References==
 
* Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications '''3'''. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 {{ZBL|0919.05001}}
 
* Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications '''3'''. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 {{ZBL|0919.05001}}
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* Golumbic, Martin Charles; Trenk, Ann N. ''Tolerance graphs'' Cambridge Studies in Advanced Mathematics '''89''' Cambridge University Press (2004) ISBN 0-521-82758-2 {{ZBL|1091.05001}}
  
 
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Revision as of 14:16, 10 January 2016

2020 Mathematics Subject Classification: Primary: 05C [MSN][ZBL]

A finite unoriented graph $G=(V,E)$ with a weight function $w : V \rightarrow \mathbf{R}$ and a threshold value $T$ such that a set $S$ of vertices is independent (has no edges) if and only if $\sum_{v \in S} w(v) < T$. The following properties characterise threshold graphs:

  • They have Dilworth number $1$: for any two vertices $x,y$, one of the neighbourhoods $N(x)$, $N(y)$ contains the other.
  • They have no induced subgraph of the form $K_{2,2}$ (complete bipartite on $2+2$ points) , $C_4$ (cycle of length $4$) or $P_4$ (path of length $4$).
  • They are the comparability graphs of threshold orders.

A graph $G$ is a threshold graph if and only if the graph complement $\bar G$ is a threshold graph.

There is a polynomial time algorithm for computing the Dilworth number of a finite graph and so for recognising a threshold graph.

References

  • Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications 3. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 Zbl 0919.05001
  • Golumbic, Martin Charles; Trenk, Ann N. Tolerance graphs Cambridge Studies in Advanced Mathematics 89 Cambridge University Press (2004) ISBN 0-521-82758-2 Zbl 1091.05001
How to Cite This Entry:
Threshold graph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Threshold_graph&oldid=37428
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