Computational geometry: an introduction
Computational geometry: an introduction
The complexity of minimizing wire lengths in VLSI layouts
Information Processing Letters
On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
Automatic graph drawing and readability of diagrams
IEEE Transactions on Systems, Man and Cybernetics
The logic engine and the realization problem for nearest neighbor graphs
Theoretical Computer Science - Special issue on theoretical computer science in Australia and New Zealand
The techniques of Komolgorov and Bardzin for three-dimensional orthogonal graph drawings
Information Processing Letters
Three-dimensional orthogonal graph drawing algorithms
Discrete Applied Mathematics
On the Computational Complexity of Upward and Rectilinear Planarity Testing
SIAM Journal on Computing
Embedding problems for paths with direction constrained edges
Theoretical Computer Science
Nice drawings of graphs are computationally hard
Selected Contributions from the on 7th Interdisciplinary Workshop on Informatics and Psychology: Visualization in Human-Computer Interaction
Two Algorithms for Three Dimensional Orthogonal Graph Drawing
GD '96 Proceedings of the Symposium on Graph Drawing
Optimal three-dimensional orthogonal graph drawing in the general position model
Theoretical Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
A note on 3D orthogonal drawings with direction constrained edges
Information Processing Letters
Graph Theory With Applications
Graph Theory With Applications
A note on 3D orthogonal graph drawing
Discrete Applied Mathematics
The three dimensional logic engine
GD'04 Proceedings of the 12th international conference on Graph Drawing
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In this paper we consider the problem of finding three-dimensional orthogonal drawings of maximum degree six graphs from the computational complexity perspective. We introduce a 3SAT reduction framework that can be used to prove the NP-hardness of finding three-dimensional orthogonal drawings with specific constraints. By using the framework we show that, given a three-dimensional orthogonal shape of a graph (a description of the sequence of axis-parallel segments of each edge), finding the coordinates for nodes and bends such that the drawing has no intersection is NP-complete. Conversely, we show that if node coordinates are fixed, finding a shape for the edges that is compatible with a non-intersecting drawing is a feasible problem, which becomes NP-complete if 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 drawing with at most two bends per edge and of characterizing orthogonal shapes admitting an orthogonal drawing without intersections.