Journal of Algorithms
On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
Reconstructing sets of orthogonal line segments in the plane
Discrete Mathematics
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Discrete & Computational Geometry
Hamiltonian orthogeodesic alternating paths
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Orthogeodesic point-set embedding of trees
GD'11 Proceedings of the 19th international conference on Graph Drawing
On the hardness of point-set embeddability
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Drawing graphs with vertices at specified positions and crossings at large angles
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Hamiltonian orthogeodesic alternating paths
Journal of Discrete Algorithms
Two-Sided boundary labeling with adjacent sides
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Bend-optimal orthogonal graph drawing in the general position model
Computational Geometry: Theory and Applications
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In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is $\mathcal{NP}$-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is $\mathcal{NP}$-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.