Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
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
Treewidth: Algorithmoc Techniques and Results
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Straight Line Embeddings of Planar Graphs on Point Sets
Proceedings of the 8th Canadian Conference on Computational Geometry
On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
Graph minors. XVIII. tree-decompositions and well-quasi-ordering
Journal of Combinatorial Theory Series B
On the number of crossing-free matchings, (cycles, and partitions)
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Drawing colored graphs on colored points
Theoretical Computer Science
Cutwidth I: A linear time fixed parameter algorithm
Journal of Algorithms
Cutwidth II: Algorithms for partial w-trees of bounded degree
Journal of Algorithms
Upward point-set embeddability
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
Improved algorithms for the point-set embeddability problem for plane 3-trees
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Point-set embeddings of plane 3-trees
Computational Geometry: Theory and Applications
On the hardness of point-set embeddability
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
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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In this paper, we study the point-set-embeddability-problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We also prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set-embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs.