Orthogonal graph drawing with inflexible edges

Abstract 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 [1] . On the other hand, FlexDraw can be solved efficiently if flex ( e ) ≥ 1 [2] and is trivial if flex ( e ) ≥ 2 [3] 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 e > 0 FlexDraw is NP-complete for instances with O ( n e ) inflexible edges with pairwise distance Ω ( n 1 − e ) (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 ( 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.