Geodesic
Stable sets for $(P₆,K_{2,3})$-free graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 387-401.

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The Maximum Stable Set (MS) problem is a well known NP-hard problem. However different graph classes for which MS can be efficiently solved have been detected and the augmenting graph technique seems to be a fruitful tool to this aim. In this paper we apply a recent characterization of minimal augmenting graphs [22] to prove that MS can be solved for (P₆,K_2,3)-free graphs in polynomial time, extending some known results.
Keywords: graph algorithms, stable sets, P₆-free graphs 
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Mosca, Raffaele. Stable sets for $(P₆,K_{2,3})$-free graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 387-401. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a0/

[1] V.E. Alekseev, On the local restriction effect on the complexity of finding the graph independence number in: Combinatorial-algebraic Methods in Applied Mathematics, (Gorkiy University Press, Gorkiy, 1983) 3-13 (in Russian).

[2] V.E. Alekseev, A polynomial algorithm for finding largest independent sets in fork-free graphs, Discrete Anal. Oper. Res., Ser. 1, 6 (1999) 3-19 (in Russian) (see also [3] for the English version).

[3] V.E. Alekseev, A polynomial algorithm for finding largest independent sets in fork-free graphs, Discrete Applied Math. 135 (2004) 3-16, doi: 10.1016/S0166-218X(02)00290-1.

[4] V.E. Alekseev, On easy and hard hereditary classes of graphs with respect to the independent set problem, Discrete Applied Math. 132 (2004) 17-26, doi: 10.1016/S0166-218X(03)00387-1.

[5] V.E. Alekseev and V.V. Lozin, Augmenting graphs for independent sets, Discrete Applied Math. 145 (2004) 3-10, doi: 10.1016/j.dam.2003.09.003.

[6] G. Bacsó and Zs. Tuza, A characterization of graphs without long induced paths, J. Graph Theory 14 (1990) 455-464, doi: 10.1002/jgt.3190140409.

[7] A. Brandstädt and Chính T. Hoáng, On clique separators, nearly chordal graphs and the Maximum Weight Stable Set problem, Theoretical Computer Science 389 (2007)) 295-306, doi: 10.1016/j.tcs.2007.09.031.

[8] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A Survey, SIAM Monographs on Discrete Math. Appl. (vol. 3, SIAM, Philadelphia, 1999).

[9] A. Brandstädt, T. Klembt and S. Mahfud, P₆- and Triangle-Free Graphs Revisited: Structure and Bounded Clique-Width, Discrete Math. and Theoretical Computer Science 8 (2006) 173-188.

[10] J. Dong, On the i-diameter of i-center in a graph without long induced paths, J. Graph Theory 30 (1999) 235-241, doi: 10.1002/(SICI)1097-0118(199903)30:3235::AID-JGT8>3.0.CO;2-C.

[11] M. Farber, On diameters and radii of bridged graphs, Discrete Math. 73 (1989) 249-260, doi: 10.1016/0012-365X(89)90268-9.

[12] J.-L. Fouquet, V. Giakoumakis and J.-M. Vanherpe, Bipartite graphs totally decomposable by canonical decomposition, International J. Foundations of Computer Science 10 (1999) 513-533, doi: 10.1142/S0129054199000368.

[13] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completness (Freeman, San Francisco, CA, 1979).

[14] M.U. Gerber, A. Hertz and V.V. Lozin, Stable sets in two subclasses of banner-free graphs, Discrete Applied Math. 132 (2004) 121-136, doi: 10.1016/S0166-218X(03)00395-0.

[15] M.U. Gerber and V.V. Lozin, On the stable set problem in special P₅-free graphs, Discrete Applied Math. 125 (2003) 215-224, doi: 10.1016/S0166-218X(01)00321-3.

[16] V. Giakoumakis and J.-M. Vanherpe, Linear time recognition and optimization for weak-bisplit graphs, bi-cographs and bipartite P₆-free graphs, International J. Foundations of Computer Science 14 (2003) 107-136, doi: 10.1142/S0129054103001625.

[17] A. Hertz and V.V. Lozin The maximum independent set problem and augmenting graphs, Graph Theory and Combinatorial Optimization, GERAD 25th Anniv., Springer, New York (2005) 69-99.

[18] P. van't Hof and D. Paulusma, A new characterization of P₆-free graphs, Discrete Applied Math. 158 (2010) 731-740, doi: 10.1016/j.dam.2008.08.025.

[19] J. Liu, Y. Peng and C. Zhao, Characterization of P₆-free graphs, Discrete Applied Math. 155 (2007) 1038-1043, doi: 10.1016/j.dam.2006.11.005.

[20] J. Liu and H. Zhou, Dominating subgraphs in graphs with some forbidden structure, Discrete Math. 135 (1994) 163-168, doi: 10.1016/0012-365X(93)E0111-G.

[21] V.V. Lozin and M. Milanič, A polynomial algorithm to find an independent set of maximum weight in a fork-free graph, J. Discrete Algorithms 6 (2008) 595-604, doi: 10.1016/j.jda.2008.04.001.

[22] V.V. Lozin and M. Milanič, On finding augmenting graphs, Discrete Applied Math. 156 (2008) 2517-2529, doi: 10.1016/j.dam.2008.03.008.

[23] V.V. Lozin and R. Mosca, Independent sets in extensions of 2K₂-free graphs, Discrete Applied Math. 146 (2005) 74-80, doi: 10.1016/j.dam.2004.07.006.

[24] V.V. Lozin and D. Rautenbach, Some results on graphs without long induced paths, Information Processing Letters 88 (2003) 167-171, doi: 10.1016/j.ipl.2003.07.004.

[25] G.J. Minty, On maximal independent sets of vertices in claw-free graphs, J. Combin. Theory (B) 28 (1980) 284-304, doi: 10.1016/0095-8956(80)90074-X.

[26] R. Mosca, Stable sets in certain P₆-free graphs, Discrete Applied Math. 92 (1999) 177-191, doi: 10.1016/S0166-218X(99)00046-3.

[27] R. Mosca, Some observations on maximum weight stable sets in certain P₅-free graphs, European J. Operational Research 184 (2008) 849-859, doi: 10.1016/j.ejor.2006.12.011.

[28] R. Mosca, Independent sets in (P₆,diamond)-free graphs, Discrete Math. and Theoretical Computer Science 11:1 (2009) 125-140.

[29] O.J. Murphy, Computing independent sets in graphs with large girth, Discrete Applied Math. 35 (1992) 167-170, doi: 10.1016/0166-218X(92)90041-8.

[30] N. Sbihi, Algorithme de recherche d'un stable de cardinalité maximum dans un graphe sans étoile, Discrete Math. 29 (1980) 53-76, doi: 10.1016/0012-365X(90)90287-R.