Combinatorial optimization with 2-joins

  • Authors:

  • Nicolas Trotignon;Kristina Vušković

  • Affiliations:

  • CNRS, LIP - ENS Lyon, France;School of Computing, University of Leeds, Leeds LS2 9JT, UK and Faculty of Computer Science, Union University, Knez Mihailova 6/VI, 11000 Belgrade, Serbia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.