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A Characterization of Comparability Graphs and of Interval Graphs

  • P. C. Gilmore (a1) and A. J. Hoffman
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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.

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References
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1. Benzer, S., On the topology of the genetic fine structure, Proc. Natl. Acad. Sci. U.S., 45 (1959), 16071620.
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), 13701371.
3. Gilmore, P. C. and Hoffman, A. J., Characterizations of Comparability and Interval Graphs, Abstract, Internat. Congress Mathematicians (Stockholm, 1962), p. 29.
4. Hajos, G., Über eine Art von Graphen, Intern. Math. Nachr., 11 (1957), Sondernummer 65.
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), 4564.
6. Wolk, E. S., The comparability graph of a tree, Proc. Am. Math. Soc, 13 (1962), 789795.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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