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)(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⩽∞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.