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

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.

Planar L-Drawings of Directed Graphs

“…The case of an R-node µ is detailed in Appendix B. 4. For each node v of skel(µ), we have to consider the virtual edges e 1 , .…”

“…1c. It is NP-complete [4] to decide whether a multigraph has a planar embedding that allows a Kandinsky drawing with at most one bend per edge [5]. On the other hand, every simple planar graph has a Kandinsky drawing with at most one bend per edge [5].…”

“…On the other hand, every simple planar graph has a Kandinsky drawing with at most one bend per edge [5]. Bend-minimization in the Kandinsky-model is NP-complete [4] even if a planar embedding is given, but can be approximated by a factor of two [2,11]. Heuristics for drawings in the Kandinsky model with empty faces and few bends have been discussed by Bekos et al [3].…”

Coupling architectural synthesis to applied thermal engineering, constructal thermodynamics and fractal analysis: An original pedagogic method to incorporate “sustainability” into architectural education during the initial conceptual stages

“…More precisely, Kandinsky's compositions are based on the marriage of sequences of horizontal, vertical and cyclical segments placed in intuitive manner on the canvas, while they seem to follow a potential linear time algorithm [33]. That's why being inspired from Kandinsky's methodology of composition a so-called Kandinsky model in informatics and discrete mathematics is a model that allows vertices of degree > 4 [34] and is developed to identify orthogonal grid embedding's of plane graphs with the minimum number of bends, solving long-standing open problems [35].…”

Planar L-Drawings of Bimodal Graphs

“…1(d)). Bend-minimization in the Kandinsky model is NP-complete [8] and can be approximated within a factor of two [2]. Each undirected simple graph admits a Kandinsky drawing with one bend per edge [9].…”