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A Characterization of Comparability Graphs and of Interval Graphs

Published online by Cambridge University Press:  20 November 2018

P. C. Gilmore  and
A. J. Hoffman
P. C. Gilmore
Affiliation:
IBM Research Center
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Let < be a non-reflexive partial ordering defined on a set P. Let G(P, <) be the undirected graph whose vertices are the elements of P, and whose edges (a, b) connect vertices for which either a < b or b < a. A graph G with vertices P for which there exists a partial ordering < such that G = G(P, <) is called a comparability graph.

In §2 we state and prove a characterization of those graphs, finite or infinite, which are comparability graphs. Another proof of the same characterization has been given in (2), and a related question examined in (6). Our proof of the sufficiency of the characterization yields a very simple algorithm for directing all the edges of a comparability graph in such a way that the resulting graph partially orders its vertices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 539 - 548
Copyright
Copyright © Canadian Mathematical Society 1964

References

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