From Tree-Width to Clique-Width: Excluding a Unit Interval Graph

Authors:
Vadim V. Lozin
Affiliations:
DIMAP and Mathematics Institute, University of Warwick, Coventry, UK CV4 7AL
Venue:
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Year:
2008

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Abstract

From the theory of graph minors we know that the class of planar graphs is the only critical class with respect to tree-width. In the present paper, we reveal a critical class with respect to clique-width, a notion generalizing tree-width. This class is known in the literature under different names, such as unit interval, proper interval or indifference graphs, and has important applications in various fields, including molecular biology. We prove that the unit interval graphs constitute a minimal hereditary class of unbounded clique-width. As an application, we show that list coloring is fixed parameter tractable in the class of unit interval graphs.