A comparability graph is a graph whose edges can be oriented transitively. Given a comparability graph G = (V,E) and an arbitrary edge ê∈ E we explore the question whether the graph G-ê, obtained by removing the undirected edge ê, is a comparability graph as well. We define a new substructure of implication classes and present a complete mathematical characterization of all those edges.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1457,
author = {Michael Andresen},
title = {On transitive orientations of G-\^e},
journal = {Discussiones Mathematicae Graph Theory},
volume = {29},
year = {2009},
pages = {423-467},
zbl = {1193.05140},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1457}
}
Michael Andresen. On transitive orientations of G-ê. Discussiones Mathematicae Graph Theory, Tome 29 (2009) pp. 423-467. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1457/
[000] [1] H. Bräsel, Lateinische Rechtecke und Maschinenbelegung (Habilitationsschrift. Technische Universität Otto-von-Guericke Magdeburg, 1990).
[001] [2] H. Bräsel, Matrices in Shop Scheduling Problems, in: M. Morlock, C. Schwindt, N. Trautmann and J. Zimmermann, eds, Perspectives on Operations Research - Essays in Honor of Klaus Neumann (Gabler Edition Wissenschaft, Deutscher Universitätsverlag, 2006), 17-43.
[002] [3] H. Bräsel, M. Harborth, T. Tautenhahn and P. Willenius, On the set of solutions of an open shop Problem, Ann. Oper. Res. 92 (1999) 241-263, doi: 10.1023/A:1018938915709. | Zbl 0958.90035
[003] [4] A. Cournier and M. Habib, A new linear algorithm for modular decomposition, in: S. Tison ed., Trees in Algebra and Programming, CAAP '94, 19th International Colloquium 787 of Lecture Notes in Computer Science (Springer Verlag, 1994) 68-82.
[004] [5] T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18 (1967) 25-66, doi: 10.1007/BF02020961.
[005] [6] P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16 (1964) 539-548, doi: 10.4153/CJM-1964-055-5. | Zbl 0121.26003
[006] [7] M.C. Golumbic, Comparability graphs and a new matroid, J. Combin. Theory (B) 22 (1977) 68-90, doi: 10.1016/0095-8956(77)90049-1. | Zbl 0352.05023
[007] [8] M.C. Golumbic, The complexity of comparability graph recognition and coloring, Comp. 18 (1977) 199-208, doi: 10.1007/BF02253207. | Zbl 0365.05025
[008] [9] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, 1980).
[009] [10] R.M. McConnell and J.P. Spinrad, Linear-time modular decomposition and efficient transitive orientation of comparability graphs, in: Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms 5 (1994) 536-545. | Zbl 0867.05068
[010] [11] R.M. McConnell and J.P. Spinrad, Modular decomposition and transitive orientation, Discrete Math. 201 (1999) 189-241, doi: 10.1016/S0012-365X(98)00319-7. | Zbl 0933.05146
[011] [12] R.M. McConnell and J.P. Spinrad, Ordered vertex partitioning, Discrete Math. and Theor. Comp. Sci. 4 (2000) 45-60. | Zbl 0946.68101
[012] [13] M. Moerig, Modulare Dekomposition durch geordnete Partitionierung der Knotenmenge: Grundlagen und Implementierung, (Diplomarbeit, Otto-von-Guericke-Universität Magdeburg, 2006).
[013] [14] A. Natanzon, R. Shamir and R. Sharan, Complexity classification of some edge modification problems, Discrete Appl. Math. 113 (2001) 109-128, doi: 10.1016/S0166-218X(00)00391-7. | Zbl 0982.68104
[014] [15] K. Simon, Effiziente Algorithmen für perfekte Graphen (Teubner, 1992).
[015] [16] P. Willenius, Irreduzibilitätstheorie bei Shop-Scheduling-Problemen (Dissertationsschrift, Shaker Verlag, 2000).
[016] [17] M. Yannakakis, Edge deletion problems, SIAM J. Comput. 10 (2) (1981) 297-309, doi: 10.1137/0210021. | Zbl 0468.05043