Abstract
We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings. We prove that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes when restricted to such graphs. This extends previous results by Baker and Eppstein to a much broader class of graphs. We also prove that for the considered class of graphs, there are balanced separators of size \(O(\sqrt{n})\) where n is a number of vertices in the graph. On the negative side, we prove that it is intractable to recognize the graphs embeddable in the plane with at most one crossing per edge.
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Grigoriev, A., Bodlaender, H. Algorithms for Graphs Embeddable with Few Crossings per Edge. Algorithmica 49, 1–11 (2007). https://doi.org/10.1007/s00453-007-0010-x
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DOI: https://doi.org/10.1007/s00453-007-0010-x