Cutwidth: Obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2^{O(k3log k)}. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2^{O(k2 log k)} · n, where k is the optimum width and n is the number of vertices. While being slower by a log k-factor in the exponent than the fastest known algorithm, due to Thilikos, Bodlaender, and Serna [17, 18], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.