On the stable set problem in special P5-free graphs

The complexity status of the stable set problem in P5-free graphs remains an open question for a long time in spite of a lot of particular results in this direction. The purpose of the present paper is to summarize these results and to propose several new ones. In particular, we prove that the problem of finding a maximum stable set can be solved in polynomial time in the class of (P5,Km,m)-free graphs for any fixed m.

[1]  S. Poljak.  A note on stable sets and colorings of graphs , 1974 .

[2]  Vadim V. Lozin,et al.  A note on alpha-redundant vertices in graphs , 2001, Discret. Appl. Math..

[3]  Zsolt Tuza,et al.  An upper bound on the number of cliques in a graph , 1993, Networks.

[4]  Vassilis Giakoumakis,et al.  On semi-P4-sparse graphs , 1997, Discret. Math..

[5]  Frederic Maire,et al.  On graphs without P5 and P5_ , 1995, Discret. Math..

[6]  Dennis Saleh.  Zs , 2001 .

[7]  Zsolt Tuza,et al.  Graphs with no induced C4 and 2K2 , 1993, Discret. Math..

[8]  Caterina De Simone,et al.  Stability Number of Bull- and Chair-Free Graphs , 1993, Discret. Appl. Math..

[9]  Derek G. Corneil.  The complexity of generalized clique packing , 1985, Discret. Appl. Math..

[10]  Ryan B. Hayward,et al.  Murky graphs , 1990, J. Comb. Theory, Ser. B.

[11]  A. Hertz.  ON A GRAPH TRANSFORMATION THAT PRESERVES THE STABILITY NUMBER , 2000 .

[12]  George J. Minty,et al.  On maximal independent sets of vertices in claw-free graphs , 1980, J. Comb. Theory, Ser. B.

[13]  Alain Hertz,et al.  Polynomially Solvable Cases for the Maximum Stable Set Problem , 1991, Discret. Appl. Math..

[14]  Alain Hertz,et al.  On the Use of Boolean Methods for the Computation of the Stability Number , 1997, Discret. Appl. Math..

[15]  Myriam Preissmann,et al.  Linear Recognition of Pseudo-split Graphs , 1994, Discret. Appl. Math..

[16]  Raffaele Mosca,et al.  Polynomial Algorithms for the Maximum Stable Set Problem on Particular Classes of P_5-Free Graphs , 1997, Inf. Process. Lett..

[17]  Vadim V. Lozin,et al.  Stability in P5- and banner-free graphs , 2000, Eur. J. Oper. Res..

[18]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[19]  P. Hammer,et al.  Quasimonotone Boolean Functions and Bistellar Graphs , 1980 .

[20]  Peter L. Hammer,et al.  On the Stability Number of Claw-free P5-free and More General Graphs , 1999, Discret. Appl. Math..

[21]  Lorna Stewart,et al.  A Linear Recognition Algorithm for Cographs , 1985, SIAM J. Comput..

[22]  Vassilis Giakoumakis,et al.  Weighted Parameters in (P5, P5)-free Graphs , 1997, Discret. Appl. Math..

[23]  L. Lovász,et al.  Polynomial Algorithms for Perfect Graphs , 1984 .

[24]  Dominique de Werra,et al.  Four classes of perfectly orderable graphs , 1987, J. Graph Theory.