Parameterized Complexity of 1-Planarity
Michael J. Bannister, Sergio Cabello, David Eppstein
TL;DR
This paper studies the computational complexity of finding 1-planar graph drawings, providing fixed-parameter tractable algorithms for certain parameters while showing NP-completeness for others.
Contribution
It introduces fixed-parameter algorithms for 1-planarity based on vertex cover, tree-depth, and cyclomatic number, and proves NP-completeness for graphs with bounded bandwidth, pathwidth, or treewidth.
Findings
FPT algorithms for vertex cover, tree-depth, cyclomatic number
NP-completeness for bounded bandwidth, pathwidth, treewidth
Complexity varies with different graph parameters
Abstract
We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.
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