Georgia Institute of Technology College of Sciences

Chudnovsky and Seymour's structure theorem for quasi-line graphs has led to a multitude of recent results that exploit two structural operations: compositions of strips and thickenings. In this paper we prove that compositions of linear interval strips have a unique optimal strip decomposition in the absence of a specific degeneracy, and that every claw-free graph has a unique optimal antithickening, where our two definitions of optimal are chosen carefully to respect the structural foundation of the graph. Furthermore, we give algorithms to find the optimal strip decomposition in O(nm) time and find the optimal antithickening in O(m2) time. For the sake of both completeness and ease of proof, we prove stronger results in the more general setting of trigraphs. This gives a comprehensive "black box" for decomposing quasi-line graphs that is not only useful for future work but also improves the complexity of some previous algorithmic results. Joint work with Maria Chudnovsky.