A simple algorithm for the graph minor decomposition: logic meets structural graph theory

A key result of Robertson and Seymour's graph minor theory is a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface. Most algorithmic applications of graph minor theory rely on an algorithmic version of this result. However, the known algorithms for computing such graph minor decompositions heavily rely on the very long and complicated proofs of the existence of such decompositions, essentially they retrace these proofs and show that all steps are algorithmic. In this paper, we give a simple quadratic time algorithm for computing graph minor decompositions. The best previously known algorithm due to Kawarabayashi and Wollan runs in cubic time and is far more complicated. Our algorithm combines techniques from logic and structural graph theory, or more precisely, a variant of Courcelle's Theorem stating that monadic second-order logic formulas can be evaluated in linear time on graphs of bounded tree width and Robertson and Seymour's so called Weak Structure Theorem.