# Comparability graph

From Encyclopedia of Mathematics

The undirected graph $(P,E)$ on a partially ordered set $(P,{\le})$ in which two points are adjacent if they are comparable; that is, $xy$ is an edge of the graph if and only if $x < y$ or $y < x$. Comparability graphs are characterised by the property that in any odd length closed path $x_1,\ldots,x_{2n+1}$ with $n \ge 2$ (so all $x_i,x_{i+1}$ are adjacent) there exists at least one "chord" $x_i,x_{i+2}$ (subscripts being taken in cyclic order).

#### 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

**How to Cite This Entry:**

Comparability graph.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Comparability_graph&oldid=37409