Abstract
We present the first polynomial-time algorithm to solve the Maximum Weight Independent Set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free.
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References
Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. USA 43, 842–844 (1957)
Brandstädt, A., Hoàng, C.T.: On Clique Separators, Nearly Chordal Graphs, and the Maximum Weight Stable Set Problem. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 265–275. Springer, Heidelberg (2005); Theoretical Computer Science 389, 295–306 (2007)
Brandstädt, A., Le, V.B., Mahfud, S.: New applications of clique separator decomposition for the Maximum Weight Stable Set Problem. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 505–516. Springer, Heidelberg (2005); Theoretical Computer Science 370, 229–239 (2007)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Math. Appl., vol. 3. SIAM, Philadelphia (1999)
Chudnovsky, M., Seymour, P.: The roots of the independence polynomial of a clawfree graph. J. Combin. Theory Ser. B 97(3), 350–357 (2007)
Chudnovsky, M., Seymour, P.: Clawfree graphs I – Orientable prismatic graphs. J. Combin. Theory Ser. B 97, 867–901 (2007)
Chudnovsky, M., Seymour, P.: Clawfree graphs II – Nonorientable prismatic graphs. J. Combin. Theory Ser. B 98, 249–290 (2008)
Chudnovsky, M., Seymour, P.: Clawfree graphs III – Circular interval graphs. J. Combin. Theory Ser. B (to appear)
Chudnovsky, M., Seymour, P.: Clawfree graphs IV – Decomposition theorem. J. Combin. Theory Ser. B (to appear)
Chudnovsky, M., Seymour, P.: Clawfree graphs V – Global structure. J. Combin. Theory Ser. B (to appear)
Chudnovsky, M., Seymour, P.: Clawfree graphs VI – The structure of quasi-line graphs (submitted)
Chudnovsky, M., Seymour, P.: Clawfree graphs VII – Coloring claw-free graphs (submitted)
Corneil, D.G., Lerchs, H., Stewart, L.: Complement reducible graphs. Discrete Appl. Math. 3, 163–174 (1981)
De Simone, C.: On the vertex packing problem. Graphs and Combinatorics 9, 19–30 (1993)
Edmonds, J.: Paths, trees and flowers. Canad. J. of Mathematics 17, 449–467 (1965)
Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Nat. Bur. Standards Sect. B 69B, 125–130 (1965)
Faudree, R., Flandrin, E., Ryjáček, Z.: Claw-free graphs – a survey. Discrete Math. 164, 87–147 (1997)
Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen 1975), pp. 211–226 (1975); Congressus Numerantium No. XV, Utilitas Math., Winnipeg, Man. (1976)
Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Computing 1, 180–187 (1972)
Gerber, M.U., Lozin, V.V.: Robust algorithms for the stable set problem. Graphs and Combinatorics 19, 347–356 (2003)
Golumbic, M.C.: Algorithmic graph theory and perfect graphs, 2nd edn. Annals of Discrete Mathematics, vol. 57. Elsevier Science B.V, Amsterdam (2004)
Lovász, L., Plummer, M.D.: Matching theory. Annals of Discrete Mathematics 29 (1986)
Lozin, V.V.: Stability in P 5- and banner-free graphs. European Journal of Operational Research 125, 292–297 (2000)
Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Combinatorial Theory, Ser. B 28, 284–304 (1980)
Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weight stable set of a claw-free graph. J. Oper. Res. Soc. Japan 44, 194–204 (2001)
Olariu, S.: The strong perfect graph conjecture for pan-free graphs. J. Combin. Th (B) 47, 187–191 (1989)
Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29, 53–76 (1980)
Spinrad, J.P.: Efficient Graph Representations, Fields Institute Monographs 19. American Mathematical Society, Providence (2003)
Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55, 221–232 (1985)
Whitesides, S.H.: A method for solving certain graph recognition and optimization problems, with applications to perfect graphs. In: Berge, C., Chvátal, V. (eds.) Topics on perfect graphs. North-Holland, Amsterdam (1984)
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Brandstädt, A., Klembt, T., Lozin, V.V., Mosca, R. (2008). Independent Sets of Maximum Weight in Apple-Free Graphs. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_74
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DOI: https://doi.org/10.1007/978-3-540-92182-0_74
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