Realizability of Delaunay triangulations
Information Processing Letters
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ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Proximity Drawings of Outerplanar Graphs
GD '96 Proceedings of the Symposium on Graph Drawing
Proximity Drawability: a Survey
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
Computational Geometry: Theory and Applications
On the area requirements of Euclidean minimum spanning trees
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
The approximate rectangle of influence drawability problem
GD'12 Proceedings of the 20th international conference on Graph Drawing
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We introduce and study a generalization of the well-known region of influence proximity drawings, called (ε1 ,ε2 )-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε1 ≥0 and ε2 ≥0, an (ε1 ,ε2 )-proximity drawing of a graph is a planar straight-line drawing $#915; such that: (i) for every pair of adjacent vertices u,v, their proximity region "shrunk" by the multiplicative factor $\frac{1}{1+\varepsilon_1}$ does not contain any vertices of $#915;; (ii) for every pair of non-adjacent vertices u,v, their proximity region "blown-up" by the factor (1+ε2 ) contains some vertices of $#915; other than u and v. We show that by using this generalization, we can significantly enlarge the family of the representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε1 and ε2 . We also study the extremal case of (0,ε2 )-proximity drawings, which generalizes the well-known weak proximity drawing model.