On embedding a graph in the grid with the minimum number of bends
SIAM Journal on Computing
The Hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs
Journal of Algorithms
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
On the Computational Complexity of Upward and Rectilinear Planarity Testing
SIAM Journal on Computing
Nice Drawings for Planar Bipartite Graphs
CIAC '97 Proceedings of the Third Italian Conference on Algorithms and Complexity
On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
Convex Drawings of 3-Connected Plane Graphs
Algorithmica
Minimizing the area for planar straight-line grid drawings
GD'07 Proceedings of the 15th international conference on Graph drawing
Minimum-segment convex drawings of 3-connected cubic plane graphs
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
Point-set embeddings of plane 3-trees
Computational Geometry: Theory and Applications
Manhattan-Geodesic embedding of planar graphs
GD'09 Proceedings of the 17th international conference on Graph Drawing
The point-set embeddability problem for plane graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
Kinetic and stationary point-set embeddability for plane graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Point-Set embeddability of 2-colored trees
GD'12 Proceedings of the 20th international conference on Graph Drawing
Plane 3-trees: embeddability and approximation
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Universal point sets for planar three-trees
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S. The problem of deciding whether a plane graph admits a point-set embedding on a given set of points is NP-complete for 2-connected planar graphs, but polynomial-time solvable for outerplanar graphs and plane 3-trees. In this paper we prove that the problem remains NP-complete for 3-connected planar graphs, which settles an open question posed by Cabello (Journal of Graph Algorithms and Applications, 10(2), 2000). We then show that the constraint of convexity makes the problem easier for klee graphs, which is a subclass of 3-connected planar graphs. We give a polynomial-time algorithm to decide whether a klee graph with exactly three outer vertices admits a convex point-set embedding on a given set of points and compute such an embedding if one exists.