Abstract
Let G be a group and H a subgroup of G. It is shown that there exists a partially ordered set (X, ⩽) such that G is isomorphic to the group of all automorphisms of the comparability graph of (X, ⩽) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (X, ⩽). There also exists a partially ordered set (Y, ⩽) such that G is isomorphic to the group of all automorphisms of the covering graph of (Y, ⩽) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (Y, ⩽). In this representation X and Y can be taken to be finite if G is finite and of the same cardinality as G if G is infinite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
LászlóBabai (1981) On the abstract group of automorphisms, in Combinatorics (ed. H. N. V.Temperley), Cambridge Univ. Press, Cambridge, pp. 1–40.
GarrettBirkhoff (1946) Sobre los grupos de automorfismos, Revista Unión Mat. Argentina 11, 155–157.
DavidKelly (1985) Comparability graphs, in Graphs and Order (ed. I.Rival), D. Reidel, Dordrecht, pp. 3–40.
IvanRival (1985) The diagram, in Graphs and Order (ed. I.Rival), D. Reidel, Dordrecht, pp. 103–133.
Author information
Authors and Affiliations
Additional information
Communicated by D. Duffus
Rights and permissions
About this article
Cite this article
Behrendt, G. Automorphism groups of comparability and covering graphs. Order 7, 5–9 (1990). https://doi.org/10.1007/BF00383168
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00383168