Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-12T20:25:14.056Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Benzer, S., On the topology of the genetic fine structure, Proc. Natl. Acad. Sci. U.S., 45 (1959), 1607–1620.Google Scholar
2. Ghouilà-Houri, A., Caractérisation des graphes nonorientês dont on peut orienter les aretes de manière à obtenir le graphe d'une relation d'ordre, C. R. Acad. Sci. Paris, 254 (1962), 1370–1371.Google Scholar
3. Gilmore, P. C. and Hoffman, A. J., Characterizations of Comparability and Interval Graphs, Abstract, Internat. Congress Mathematicians (Stockholm, 1962), p. 29.Google Scholar
4. Hajos, G., Über eine Art von Graphen, Intern. Math. Nachr., 11 (1957), Sondernummer 65.Google Scholar
5. Lekkerkerker, C. G. and J. Ch. Bohland, Representation of a finite graph by a set of intervals in the real line, Fund. Math., 51 (1962), 45–64.Google Scholar
6. Wolk, E. S., The comparability graph of a tree, Proc. Am. Math. Soc, 13 (1962), 789–795.Google Scholar