Abstract
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^3log 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^2log 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.
Highlights
The cutwidth of a graph is defined as the minimum possible width of a linear ordering of its vertices, where the width of an ordering σ is the maximum, among all the prefixes of σ, of the number of edges that have exactly one vertex in a prefix
Our main result concerns the sizes of obstructions for cutwidth
The above result immediately gives the same upper bound on the sizes of graphs from the minimal obstruction sets Lk as they satisfy the prerequisites of Theorem 1
Summary
The cutwidth of a graph is defined as the minimum possible width of a linear ordering of its vertices, where the width of an ordering σ is the maximum, among all the prefixes of σ , of the number of edges that have exactly one vertex in a prefix. Thilikos et al [23,24] proposed a fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2O(k2) · n, where k is the optimum width and n is the number of vertices. Their approach is to first compute the pathwidth of the input graph, which is never larger than the cutwidth. They borrow the technique of typical sequences of Bodlaender and Kloks [3], which was introduced for a similar reason, but for pathwidth and treewidth instead of cutwidth
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