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 \(2^{{O}(k^3\log k)}\). As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time \(2^{{O}(k^2\log k)}\cdot n\), 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, given by Thilikos et al. (J Algorithms 56(1):1–24, 2005; J Algorithms 56(1):25–49, 2005), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.