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Discussiones Mathematicae Graph Theory 32(3) (2012) 387-401
DOI: https://doi.org/10.7151/dmgt.1598

Stable Sets for (P₆, K_2,3)-free Graphs

Raffaele Mosca Dipartimento di Scienze Universitá degli Studi "G. D'Annunzio" Pescara, Italy

Abstract

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

2010 Mathematics Subject Classification: 05C69, 05C85.

References

[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)30:3<235::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.

Received 13 May 2010
Revised 30 June 2011
Accepted 4 July 2011