A Simpler Algorithm and Shorter Proof for the Graph Minor Decomposition

At the core of the Robertson-Seymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation, as well as give a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed. The theorem has since found numerous applications, both in graph theory and theoretical computer science. The original proof runs more than 400 pages and the techniques used are highly non-trivial. In this paper, we give a simplified algorithm for finding the decomposition based on a new constructive proof of the decomposition theorem for graphs excluding a fixed minor H. The new proof is both dramatically simpler and shorter, making these results and techniques more accessible. The algorithm runs in time O(n(3)), as does the original algorithm of Robertson and Seymour. Moreover, our proof gives an explicit bound on the constants in the O notation, whereas the original proof of Robertson and Seymour does not.