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

TL;DR

This paper proves that computing minimal-bend orthogonal grid-embeddings of plane graphs in the Kandinsky model is NP-complete, but also provides efficient algorithms for specific graph classes and a subexponential solution for general cases.

Contribution

It establishes NP-completeness for the problem and introduces efficient algorithms for restricted graph classes and a subexponential algorithm for general graphs.

Findings

NP-completeness of minimal-bend orthogonal embeddings in the Kandinsky model

Efficient algorithms for graphs with bounded branch width

Subexponential algorithm for general plane graphs

Abstract

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.