Abstract
The only available combinatorial algorithm for the minimum weighted clique cover (mwcc) in claw-free perfect graphs is due to Hsu and Nemhauser [10] and dates back to 1984. More recently, Chudnovsky and Seymour [3] introduced a composition operation, strip-composition, in order to define their structural results for claw-free graphs; however, this composition operation is general and applies to non-claw-free graphs as well. In this paper, we show that a mwcc of a perfect strip-composed graph, with the basic graphs belonging to a class \({\cal G}\), can be found in polynomial time, provided that the mwcc problem can be solved on \({\cal G}\) in polynomial time. We also design a new, more efficient, combinatorial algorithm for the mwcc problem on strip-composed claw-free perfect graphs.
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References
Burlet, M., Maffray, F., Trotignon, N.: Odd pairs of cliques. In: Proc. of a Conference in Memory of Claude Berge, pp. 85–95. Birkhäuser (2007)
Chudnovsky, M., Seymour, P.: Claw-free graphs. VII. Quasi-line graphs. J. Combin. Theory, Ser. B (to appear)
Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. London Math. Soc. Lecture Note Ser. 327, 153–171 (2005)
Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P.: Circular one matrices and the stable set polytope of quasi-line graphs. Combinatorica 28(1), 45–67 (2008)
Faenza, Y., Oriolo, G., Stauffer, G.: An algorithmic decomposition of claw-free graphs leading to an O(n 3)-algorithm for the weighted stable set problem. In: Randall, D. (ed.) Proc. 22nd SODA, San Francisco, CA, pp. 630–646 (2011)
Gabow, H.: Data structures for weighted matching and nearest common ancestors with linking. In: Proc. 1st SODA, San Francisco, CA, pp. 434–443 (1990)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)
Hermelin, D., Mnich, M., Van Leeuwen, E., Woeginger, G.: Domination When the Stars Are Out. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 462–473. Springer, Heidelberg (2011)
Hsu, W., Nemhauser, G.: Algorithms for minimum covering by cliques and maximum clique in claw-free perfect graphs. Discrete Math. 37, 181–191 (1981)
Hsu, W., Nemhauser, G.: Algorithms for maximum weight cliques, minimum weighted clique covers and cardinality colorings of claw-free perfect graphs. Ann. Discrete Math. 21, 317–329 (1984)
Minty, G.: On maximal independent sets of vertices in claw-free graphs. J. Combin. Theory, Ser. B 28(3), 284–304 (1980)
Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weighted stable set of a claw-free graph. J. Oper. Res. Soc. Japan 44(2), 194–204 (2001)
Nobili, P., Sassano, A.: A reduction algorithm for the weighted stable set problem in claw-free graphs. In: Proc. 10th CTW, Frascati, Italy, pp. 223–226 (2011)
Oriolo, G., Pietropaoli, U., Stauffer, G.: A New Algorithm for the Maximum Weighted Stable Set Problem in Claw-Free Graphs. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 77–96. Springer, Heidelberg (2008)
Schrijver, A.: Combinatorial Optimization. In: Polyhedra and Efficiency (3 volumes), Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
Trotter, L.: Line perfect graphs. Math. Program. 12, 255–259 (1977)
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Bonomo, F., Oriolo, G., Snels, C. (2012). Minimum Weighted Clique Cover on Strip-Composed Perfect Graphs. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_6
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DOI: https://doi.org/10.1007/978-3-642-34611-8_6
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