2024/06/27 by Hliněný, Petr, Khazaliya, Liana
#68R10 #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2406.18933
Crossing Number is a celebrated problem in graph drawing. It is known to be NP-complete since 1980s, and fairly involved techniques were already required to show its fixed-parameter tractability when parameterized by the vertex cover number. In this paper we prove that computing exactly the crossing number is NP-hard even for graphs of path-width 12 (and as a result, even of tree-width 9). Thus, while tree-width and path-width have been very successful tools in many graph algorithm scenarios, our result shows that general crossing number computations unlikely (under P!=NP) could be successfully tackled using bounded width of graph decompositions, which has been a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.