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 (which allows 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.
@article{arxiv.1405.2300,
title = {Complexity of Higher-Degree Orthogonal Graph Embedding in the Kandinsky Model},
author = {Thomas Bläsius and Guido Brückner and Ignaz Rutter},
journal= {arXiv preprint arXiv:1405.2300},
year = {2014}
}