Cutwidth: Obstructions and Algorithmic Aspects

Archontia C. Giannopoulou, Michał Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, Marcin Wrochna
2018 Algorithmica  
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
more » ... tion 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) · 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; Algorithmica (2019) 81:557-588 J Algorithms 56 (1) : 2005), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.
doi:10.1007/s00453-018-0424-7 fatcat:gla2zavswrdqzmglnloo5kmovq