Complexity of Higher-Degree Orthogonal Graph Embedding in the Kandinsky Model

We show that finding orthogonal grid embeddings of plane graphs (planar with fixed combinatorial embedding) with the minimum number of bends in the so-called Kandinsky model (allowing vertices of degree > 4) is NP-complete, thus solving a long-standing open problem. On the positive side, we give an efficient algorithm for several restricted variants, such as graphs of bounded branch width and a subexponential exact algorithm for general plane graphs.