Orthogonal graph drawing with inflexible edges

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)flex(e), its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge e   has at most flex(e)flex(e) bends. It is known that FlexDraw is NP-hard if flex(e)=0flex(e)=0 for every edge e [1]. On the other hand, FlexDraw can be solved efficiently if flex(e)≥1flex(e)≥1[2] and is trivial if flex(e)≥2flex(e)≥2[3] for every edge e.To close the gap between the NP-hardness for flex(e)=0flex(e)=0 and the efficient algorithm for flex(e)≥1flex(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>0ε>0FlexDraw is NP-complete for instances with O(nε)O(nε) inflexible edges with pairwise distance Ω(n1−ε)Ω(n1−ε) (including the case where they induce a matching), where n   denotes the number of vertices in the graph. On the other hand, we give an FPT-algorithm with running time O(2k⋅n⋅Tflow(n))O(2k⋅n⋅Tflow(n)), where Tflow(n)Tflow(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.