On the maximum independent set problem in subclasses of subcubic graphs

It is known that the maximum independent set problem is NP-complete for subcubic graphs, i.e. graphs of vertex degree at most 3. Moreover, the problem is NP-complete for 3-regular Hamiltonian graphs and for H-free subcubic graphs whenever H contains a connected component which is not a tree with at most 3 leaves. We show that if every connected component of H is a tree with at most 3 leaves and at most 7 vertices, then the problem can be solved for H-free subcubic graphs in polynomial time. We also strengthen the NP-completeness of the problem on 3-regular Hamiltonian graphs by showing that the problem is APX-complete in this class.

[1]  Karpinski Marek,et al.  Efficient Amplifiers and Bounded Degree Optimization , 2001 .

[2]  Klaudia Frankfurter.  Computers And Intractability A Guide To The Theory Of Np Completeness , 2016 .

[3]  Vadim V. Lozin,et al.  Maximum regular induced subgraphs in 2 P3-free graphs , 2012, Theor. Comput. Sci..

[4]  Vadim V. Lozin,et al.  Graphs Without Large Apples and the Maximum Weight Independent Set Problem , 2014, Graphs Comb..

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

[6]  Marek Karpinski,et al.  Improved Approximation Lower Bounds on Small Occurrence Optimization , 2003, Electron. Colloquium Comput. Complex..

[7]  Dimitrios M. Thilikos,et al.  Treewidth for Graphs with Small Chordality , 1997, Discret. Appl. Math..

[8]  Herbert Fleischner,et al.  Maximum independent sets in 3- and 4-regular Hamiltonian graphs , 2010, Discret. Math..

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

[10]  Vadim V. Lozin,et al.  Maximum independent sets in subclasses of P5-free graphs , 2009, Inf. Process. Lett..

[11]  Alain Hertz,et al.  P5-free augmenting graphs and the maximum stable set problem , 2002, Discret. Appl. Math..

[12]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  R. Steele.  Optimization , 2005 .

[14]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[15]  Raffaele Mosca.  Some results on maximum stable sets in certain P5-free graphs , 2003, Discret. Appl. Math..

[16]  Owen J. Murphy,et al.  Computing independent sets in graphs with large girth , 1992, Discret. Appl. Math..

[17]  Frédéric Maffray,et al.  Stable sets in k-colorable P5-free graphs , 2009, Inf. Process. Lett..

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

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