Proximity Drawings Research Articles

Approximate proximity drawings

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 Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region “shrunk” by the multiplicative factor 11+ε1 does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u, v, their proximity region “expanded” by the factor (1+ε2) contains some vertices of Γ other than u and v. In particular, the locations of the vertices in such a drawing do not always completely determine which edges must be present/absent, giving us some freedom of choice. We show that this generalization significantly enlarges the family of 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 generalize the well-known weak proximity drawing paradigm.

The strength of weak proximity

This paper studies weak proximity drawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximity drawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment ( p , q ) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximity drawing in which such segments must appear in the drawing. Most previously studied proximity regions are associated with a parameter β, 0 ⩽ β ⩽ ∞ . For fixed β, weak β-drawability is at least as expressive as strong β-drawability, as a strong β-drawing is also a weak one. We give examples of graph families and β values where the two notions coincide, and a situation in which it is NP-hard to determine weak β-drawability. On the other hand, we give situations where weak proximity significantly increases the expressive power of β-drawability: we show that every graph has, for all sufficiently small β, a weak β-proximity drawing that is computable in linear time, and we show that every tree has, for every β less than 2, a weak β-drawing that is computable in linear time.

Proximity drawings in polynomial area and volume

We introduce a novel technique for drawing proximity graphs in polynomial area and volume. Previously known algorithms produce representations whose size increases exponentially with the size of the graph. This holds even when we restrict ourselves to binary trees. Our method is quite general and yields the first algorithms to construct (a) polynomial area weak Gabriel drawings of ternary trees, (b) polynomial area weak β-proximity drawing of binary trees for any 0⩽ β<∞, and (c) polynomial volume weak Gabriel drawings of unbounded degree trees. Notice that, in general, the above graphs do not admit a strong proximity drawing. Finally, we give evidence of the effectiveness of our technique by showing that a class of graph requiring exponential area even for weak Gabriel drawings, admits a linear-volume strong β- proximity drawing and a relative neighborhood drawing. All described algorithms run in linear time.

Proximity drawings in polynomial area and volume*1

We introduce a novel technique for drawing proximity graphs in polynomial area and volume. Previously known algorithms produce representations whose size increases exponentially with the size of the graph. This holds even when we restrict ourselves to binary trees. Our method is quite general and yields the first algorithms to construct (a) polynomial area weak Gabriel drawings of ternary trees , (b) polynomial area weak β -proximity drawing of binary trees for any 0⩽ β <∞, and (c) polynomial volume weak Gabriel drawings of unbounded degree trees. Notice that, in general, the above graphs do not admit a strong proximity drawing. Finally, we give evidence of the effectiveness of our technique by showing that a class of graph requiring exponential area even for weak Gabriel drawings, admits a linear-volume strong β - proximity drawing and a relative neighborhood drawing. All described algorithms run in linear time.