<p>A book source for information about problems of this type is: Graph Classes: A Survey, A. Brandstadt, Le, and Spinrad, SIAM, 1999.</p>

CC BY-SA 3.0
2011-08-05T17:40:43.023Z
2011-08-05T17:40:43.023Z
Joseph Malkevitch
<p>I believe the answer to your question, and to most questions like this, is to be found on <a href="http://graphclasses.org/">http://graphclasses.org/</a></p>

<p>There's also a book that has much of this (including an appendix at the back with some of the main subset relations between graph classes): Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.</p>

<p>The answer to your specific question is no. The graph shown in <a href="http://commons.wikimedia.org/wiki/File:SubdividedTriangle.png">http://commons.wikimedia.org/wiki/File:SubdividedTriangle.png</a> (a central triangle with three more triangle attached to its edges) is chordal, but its complement <a href="http://en.wikipedia.org/wiki/File:Forbidden_interval_subgraph.svg">http://en.wikipedia.org/wiki/File:Forbidden_interval_subgraph.svg</a> is not a comparability graph.</p>

CC BY-SA 3.0
2011-08-04T15:42:20.950Z
I wish I give more than one up vote for this
Tyson Williams
CC BY-SA 3.0
2011-08-04T16:18:00.673Z
Actually, I am not so sure about this website any more. Their server has an error if you try to suggest new graph classes. You are then instructed to email their webmaster at webmaster@graphclasses.org but that email address doesn't exist.
Tyson Williams
CC BY-SA 3.0
2011-08-06T15:52:10.727Z
@Tyson: perhaps try emailing Ernst De Ridder, hnridder at informatik.uni-rostock.de which used to be the address to email before the move to the new domain.
András Salamon
CC BY-SA 3.0
2011-08-12T13:25:04.423Z
Posing on half of Ernst de Ridder:
I'd like to apologize for the email problems at http://graphclasses.org. Both the feedback form and the webmaster email address should now work again. Please note that my old Rostock University address as given by Andras doesn't work anymore. I can be reached these days at hnridder at graphclasses.org. Bring on those emails with new classes! :-)
Tyson Williams
CC BY-SA 3.0
2011-08-04T15:35:16.823Z
2011-08-04T15:35:16.823Z
David Eppstein
<p>I was confused by Wikipedia's definitions of "<a href="http://en.wikipedia.org/wiki/Chordal_graph" rel="nofollow">chordal graph</a>", "<a href="http://en.wikipedia.org/wiki/Interval_graph" rel="nofollow">interval graph</a>", "<a href="http://en.wikipedia.org/wiki/String_graph" rel="nofollow">string graph</a>", "<a href="http://en.wikipedia.org/wiki/Comparability_graph" rel="nofollow">comparability graph</a>", "incomparability graph" and the complements of these.
Wikipedia says "The complement of any interval graph is a comparability graph". So is any chordal graph an incomparability graph?</p>

<p>Is there a book or survey that contains detailed description of these sorts of different perfect graphs?</p>

<reference-request><graph-theory><co.combinatorics><graph-classes>
Is any chordal graph an incomparability graph?
Peng Zhang
# Is any chordal graph an incomparability graph?

(Asked by: username_0 on 2011-08-04)

I was confused by Wikipedia's definitions of "[chordal graph](http://en.wikipedia.org/wiki/Chordal_graph)", "[interval graph](http://en.wikipedia.org/wiki/Interval_graph)", "[string graph](http://en.wikipedia.org/wiki/String_graph)", "[comparability graph](http://en.wikipedia.org/wiki/Comparability_graph)", "incomparability graph" and the complements of these.
Wikipedia says "The complement of any interval graph is a comparability graph". So is any chordal graph an incomparability graph?

Is there a book or survey that contains detailed description of these sorts of different perfect graphs?

**Tags:** reference-request, graph-theory, co.combinatorics, graph-classes

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## 2 answers

### Answer by username_2 on 2011-08-04. Score: 24 | Accepted answer

I believe the answer to your question, and to most questions like this, is to be found on <http://graphclasses.org/>

There's also a book that has much of this (including an appendix at the back with some of the main subset relations between graph classes): Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.

The answer to your specific question is no. The graph shown in <http://commons.wikimedia.org/wiki/File:SubdividedTriangle.png> (a central triangle with three more triangle attached to its edges) is chordal, but its complement <http://en.wikipedia.org/wiki/File:Forbidden_interval_subgraph.svg> is not a comparability graph.

> **Comments:**
> **username_3**: I wish I give more than one up vote for this  
> **username_3**: Actually, I am not so sure about this website any more. Their server has an error if you try to suggest new graph classes. You are then instructed to email their webmaster at webmaster@graphclasses.org but that email address doesn't exist.  
> **username_4**: @Tyson: perhaps try emailing Ernst De Ridder, hnridder at informatik.uni-rostock.de which used to be the address to email before the move to the new domain.  
> **username_3**: Posing on half of Ernst de Ridder: I'd like to apologize for the email problems at http://graphclasses.org. Both the feedback form and the webmaster email address should now work again. Please note that my old Rostock University address as given by Andras doesn't work anymore. I can be reached these days at hnridder at graphclasses.org. Bring on those emails with new classes! :-)  

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### Answer by username_1 on 2011-08-05. Score: 2

A book source for information about problems of this type is: Graph Classes: A Survey, A. Brandstadt, Le, and Spinrad, SIAM, 1999.


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