Stability of Hereditary Graph Classes Under Closure Operations


  • Contract grant sponsor: Czech Ministry of Education; Contract grant numbers: 1M0545; MSM 4977751301 (to Z. R., J. T., P. V.). Contract grant sponsor: Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (to M. M.).


If inline image is a subclass of the class of claw-free graphs, then inline image is said to be stable if, for any inline image, the local completion of G at any vertex is also in inline image. If inline image is a closure operation that turns a claw-free graph into a line graph by a series of local completions and inline image is stable, then inline image for any inline image. In this article, we study stability of hereditary classes of claw-free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.