We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge e a natural number flex(e), its flexibility. The problem FLEXDRAW asks whether there exists an orthogonal drawing such that each edge e has at most flex(e) bends. It is known that FLEXDRAW is NP-hard if flex(e) = 0 for every edge e [7]. On the other hand, FLEXDRAW can be solved efficiently if flex(e) ≥ 1 [2] and is trivial if flex(e) ≥ 2 [1] for every edge e. To close the gap between the NP-hardness for flex(e) = 0 and the efficient algorithm for flex(e) ≥ 1, we investigate the computational complexity of FLEXDRAW in case only few edges are inflexible (i.e., have flexibility 0). We show that for any ε > 0 FLEXDRAW is NP-complete for instances with O(n~ε) inflexible edges with pairwise distance Ω(n~(1-ε)) (including the case where they induce a matching). On the other hand, we give an FPT-algorithm with running time O(2~k·n·T_(flow)(n)), where T_(flow)(n) is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and k is the number of inflexible edges having at least one endpoint of degree 4.
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