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  • Adding one edge to planar graphs makes crossing number and 1-planarity hard
    Cabello, Sergio ; Mohar, Bojan, 1956-
    A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is ... 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.
    Vir: SIAM journal on computing. - ISSN 0097-5397 (Vol. 42, iss. 5, 2013, str. 1803-1829)
    Vrsta gradiva - članek, sestavni del
    Leto - 2013
    Jezik - angleški
    COBISS.SI-ID - 16733017

    Povezava(-e):

    http://dx.doi.org/10.1137/120872310

vir: SIAM journal on computing. - ISSN 0097-5397 (Vol. 42, iss. 5, 2013, str. 1803-1829)
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