A strengthening of Ben Rebea's lemma
Journal of Combinatorial Theory Series B
Matching Theory (North-Holland mathematics studies)
Matching Theory (North-Holland mathematics studies)
Graphs and Hypergraphs
A polynomial algorithm to find an independent set of maximum weight in a fork-free graph
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On the Stable Set Polytope of Claw-Free Graphs
COCOA 2008 Proceedings of the 2nd international conference on Combinatorial Optimization and Applications
Separating stable sets in claw-free graphs via Padberg-Rao and compact linear programs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Operations Research Letters
Triangulation and clique separator decomposition of claw-free graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Minimum weighted clique cover on strip-composed perfect graphs
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
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In this paper, we introduce two powerful graph reductions for the maximum weighted stable set (mwss) in general graphs. We show that these reductions allow to reduce the mwss in claw-free graphs to the mwss in a class of quasi-line graphs, that we call bipolar-free. For this latter class, we provide a new algorithmic decomposition theorem running in polynomial time. We then exploit this decomposition result and our reduction tools again to transform the problem to either a single matching problem or a longest path computation in an acyclic auxiliary graph (in this latter part we use some results of Pulleyblank and Shepherd [10]). Putting all the pieces together, the main contribution of this paper is a new polynomial time algorithm for the mwss in claw-free graphs. A rough analysis of the complexity of this algorithm gives a time bound of O(n6), where n is the number of vertices in the graph, and which we hope can be improved by a finer analysis. Incidentally, we prove that the mwss problem can be solved efficiently for any class of graphs that admits a "suitable" decomposition into pieces where the mwss is easy.