Abstract
This paper studies a number of problems on cycle-free partial orders and chordal comparability graphs. The dimension of a cycle-free partial order is shown to be at most 4. A linear time algorithm is presented for determining whether a chordal directed graph is transitive, which yields an O(n 2) algorithm for recognizing chordal comparability graphs. An algorithm is presented for determining whether the transitive closure of a digraph is a cycle-free partial order in O(n+m t)time, where m tis the number of edges in the transitive closure.
Similar content being viewed by others
References
A. V.Aho, J. E.Hopcroft, and J. D.Ullman (1974) The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA.
V.Bouchitte and M.Habib (1989) The Calculation of Invariants for Ordered Sets, in Algorithms and Order, ed. I.Rival, Kluwer, Dordrecht, pp. 231–279.
P.Buneman (1974) A Characterization of Rigid Circuit Graphs, Discrete Math. 9, 205–212.
D. Coppersmith and S. Winograd (1987) Matrix Multiplication via Arithmetic Progressions, Proceedings of the 19th Annual IEEE Symposium on the Foundations of Computer Science, 1–6.
B.Dushnick and E. W.Miller (1941) Partially Ordered Sets, Amer. J. Math. 63, 600–610.
D.Duffus, I.Rival and P.Winkler (1982), Minimizing Setups for Cycle-Free Ordered Sets, Proc. Amer. Math. Soc. 85, 509–513.
M. C.Golumbic (1980) Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York.
R. H.Möhring (1989) Computationally Tractable Classes of Ordered Sets, in Algorithms and Order, ed. I.Rival, Kluwer, Dordrecht, pp. 105–194.
W. R. Pulleyblank (1982) Alternating Cycle-Free Matchings, CORR Report 82-18, Department of Combinatorics and Optimization, University of Waterloo.
J. Qin and W. T. Trotter (1990) personal communication.
D. J.Rose, R. E.Tarjan, and G. S.Leuker (1976) Algorithmic Aspects of Vertex Elimination on Graphs, SIAM J. Computing 5, 266–283.
J.Spinrad (1985) On Comparability and Permutation Graphs, SIAM J. of Computing 14, 658–670.
W. T.Trotter and J. I.Moore (1977) The Dimension of Planar Posets, Journal of Combinatorial Theory Series B 22, 54–67.
R. E.Tarjan and M.Yannakakis (1984) Simple Linear Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs, SIAM J. of Computing 23, 566–579.
Author information
Authors and Affiliations
Additional information
Communicated by I. Rival
Rights and permissions
About this article
Cite this article
Ma, TH., Spinrad, J.P. Cycle-free partial orders and chordal comparability graphs. Order 8, 49–61 (1991). https://doi.org/10.1007/BF00385814
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00385814