Abstract
We call an edge e of a perfect graph G critical if G - e is imperfect and sayfurther that e is anticritical with respect to the complementary graph \( \overline G \) . We ask in which perfect graphs critical and anticritical edges occur and how to find critical and anticritical edges in perfect graphs. Finally, we study whether we can order the edges of certain perfect graphs such that deleting all the edges yields a sequence of perfect graphs ending up with a stable set.
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References
Berge, C.: Färbungen von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Zeitschrift der Martin-Luther-Universität Halle-Wittenberg (1961) 114–115
Berge, C., Duchet, P.: Strongly Perfect Graphs. In: Berge, C., Chvátal, V. (eds.): Topics on Perfect Graphs. North Holland, Amsterdam (1984) 57–61
Chvátal, V.: Star-Cutsets and Perfect Graphs. J. Combin. Theory(B) 39 (1985) 189–199
Chvátal, V., Sbihi, N.: Bull-Free Berge Graphs are Perfect. Graphs and Combinatorics 3 (1987) 127–139
Grötschel, M., Lovász, L., Schrijver, A.: The Ellipsoid Method and its Consequences in Combinatorial Optimization. Combinatorica 1 (1981) 169–197
Hayward, R.B.: Weakly Triangulated Graphs. J. Combin. Theory(B) 39 (1985) 200–209
Hayward, R.B.: Generating Weakly Triangulated Graphs. J. Graph Theory 21, No. 1 (1996) 67–69
Hayward, R.B., Hoàng, C.T., Maffray, F.: Optimizing Weakly Triangulated Graphs. Graphs and Combinatorics 5 (1989) 339–349, erratum in 6 (1990) 33–35
Hertz, A.: Slim Graphs. Graphs and Combinatorics 5 (1989) 149–157
Hoàng, C.T.: Some Properties of Minimal Imperfect Graphs. Discrete Math. 160 (1996) 165–175
Lovász, L.: Normal Hypergraphs and the Weak Perfect Graph Conjecture. Discrete Math. 2 (1972) 253–267
Meyniel, H.: On The Perfect Graph Conjecture. Discrete Math. 16 (1976) 339–342
Meyniel, H.: A New Property of Critical Imperfect Graphs and some Consequences. Europ. J. Combinatorics 8 (1987) 313–316
Olariu, S.: No Antitwins in Minimal Imperfect Graphs. J. Combin. Theory(B) 45 (1988) 255–257
Padberg, M.W.: Perfect Zero-One Matrices. Math. Programming 6 (1974) 180–196
Parthasarathy, K.R., Ravindra, G.: The Strong Perfect Graph Conjecture is True for K1,3-free Graphs. J. Combin. Theory(B) 21 (1976) 212–223
Spinrad, J., Sritharan, R.: Algorithms for WeaklyT riangulated Graphs. Discrete Appl. Math. 59 (1995) 181–191
Tucker, A.: Critical Perfect Graphs and Perfect 3-chromatic Graphs. J. Combin. Theory(B) 23 (1977) 143–149
Tucker, A.: Coloring Perfect (K4-e)-free Graphs. J. Combin. Theory(B) 42 (1987) 313–318
Wagler, A.: Critical Edges in Perfect Graphs. PhD thesis, TU Berlin (2000)
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Wagler, A. (2001). Critical and Anticritical Edges in Perfect Graphs. In: Brandstädt, A., Le, V.B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2001. Lecture Notes in Computer Science, vol 2204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45477-2_29
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DOI: https://doi.org/10.1007/3-540-45477-2_29
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