Floor-planning by graph dualization: 2-concave rectilinear modules
SIAM Journal on Computing
Continuous cartogram construction
Proceedings of the conference on Visualization '98
SIAM Journal on Computing
On topological aspects of orientations
Discrete Mathematics
Compact floor-planning via orderly spanning trees
Journal of Algorithms
Rectangular layouts and contact graphs
ACM Transactions on Algorithms (TALG)
Contact representations of planar graphs with cubes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Optimal Polygonal Representation of Planar Graphs
Algorithmica - Special Issue: Theoretical Informatics
Proportional contact representations of planar graphs
GD'11 Proceedings of the 19th international conference on Graph Drawing
Computing cartograms with optimal complexity
Proceedings of the twenty-eighth annual symposium on Computational geometry
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In a contact representation of a planar graph, vertices are represented by interior-disjoint polygons and two polygons share a non-empty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4-connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10-year old conjecture on the existence of a Hamiltonian canonical cycle in a 4-connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NP-hard to decide whether a 4-connected planar graph admits a proportional contact representation using only rectangles.