On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
The techniques of Komolgorov and Bardzin for three-dimensional orthogonal graph drawings
Information Processing Letters
Three-dimensional orthogonal graph drawing algorithms
Discrete Applied Mathematics
Embedding problems for paths with direction constrained edges
Theoretical Computer Science
Refinement of Three-Dimensional Orthogonal Graph Drawings
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
Orthogonal Drawings of Cycles in 3D Space (Extended Abstract)
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
Optimal three-dimensional orthogonal graph drawing in the general position model
Theoretical Computer Science
A note on 3D orthogonal drawings with direction constrained edges
Information Processing Letters
Graph Theory With Applications
Graph Theory With Applications
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We introduce the 3SAT reduction framework which can be used to prove the NP-hardness of finding three-dimensional orthogonal drawings with specific constraints. We use it to show that finding a drawing of a graph whose edges have a fixed shape is NP-hard. Also, it is NP-hard finding a drawing of a graph with nodes at prescribed positions when a maximum of two bends per edge is allowed. We comment on the impact of these results on the two open problems of determining whether a graph always admits a 3D orthogonal drawing with at most two bends per edge and of characterizing orthogonal shapes admitting a drawing without intersections.