Maximum weight independent sets in classes related to claw-free graphs

Abstract The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for S 1 , 1 , 3 -free graphs, and for S 1 , 2 , 2 -free graphs is unknown. We show that the MWIS problem in ( S 1 , 1 , 3 , banner)-free graphs, and in ( S 1 , 2 , 2 , bull)-free graphs can be solved in polynomial time. These results extend some known results in the literature.

[1]  Maria Chudnovsky,et al.  The Erdös-Hajnal conjecture for bull-free graphs , 2008, J. Comb. Theory, Ser. B.

[2]  Celina M. H. de Figueiredo,et al.  Optimizing Bull-Free Perfect Graphs , 2005, SIAM J. Discret. Math..

[3]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[4]  Vassilis Giakoumakis,et al.  Maximum Weight Independent Sets in hole- and co-chair-free graphs , 2012, Inf. Process. Lett..

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  T. Karthick,et al.  On atomic structure of P5-free subclasses and Maximum Weight Independent Set problem , 2014, Theor. Comput. Sci..

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

[8]  Vadim V. Lozin,et al.  Stable sets in two subclasses of banner-free graphs , 2003, Discret. Appl. Math..

[9]  Vadim V. Lozin,et al.  Independent Sets of Maximum Weight in Apple-Free Graphs , 2008, SIAM J. Discret. Math..

[10]  Daniel Lokshtanov,et al.  Independent Set in P5-Free Graphs in Polynomial Time , 2014, SODA.

[11]  Matús Mihalák,et al.  Vertex Disjoint Paths for Dispatching in Railways , 2010, ATMOS.

[12]  Yasuhiko Morimoto,et al.  Data Mining with optimized two-dimensional association rules , 2001, TODS.

[13]  Vadim V. Lozin,et al.  Independent sets in extensions of 2K2-free graphs , 2005, Discret. Appl. Math..

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

[15]  Andreas Brandstädt,et al.  Stability number of bull- and chair-free graphs revisited , 2003, Discret. Appl. Math..

[16]  Andreas Brandstädt,et al.  Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull , 2012, Graphs Comb..

[17]  Vadim V. Lozin,et al.  Finding augmenting chains in extensions of claw-free graphs , 2003, Inf. Process. Lett..

[18]  Raffaele Mosca.  Maximum weight independent sets in (P6, co-banner)-free graphs , 2013, Inf. Process. Lett..

[19]  Manu Basavaraju,et al.  Maximum weight independent sets in hole- and dart-free graphs , 2012, Discret. Appl. Math..

[20]  Stephan Olariu.  On the homogeneous representation of interval graphs , 1991, J. Graph Theory.

[21]  Robert E. Tarjan,et al.  Decomposition by clique separators , 1985, Discret. Math..

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

[23]  Martin Milanič,et al.  A polynomial algorithm to find an independent set of maximum weight in a fork-free graph , 2006, SODA '06.

[24]  Bruce Reed,et al.  Recognizing bull-free perfect graphs , 1995, Graphs Comb..

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

[26]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[27]  Vassilis Giakoumakis,et al.  Addendum to: Maximum Weight Independent Sets in hole- and co-chair-free graphs , 2015, Inf. Process. Lett..