The approximate rectangle of influence drawability problem

Abstract

We prove that all planar graphs have an open/closed (ε1, ε2)-rectangle of influence drawing for ε1 > 0 and ε2 > 0, while there are planar graphs which do not admit an open/closed (ε1, 0)-rectangle of influence drawing and planar graphs which do not admit a (0, ε2)-rectangle of influence drawing. We then show that all outerplanar graphs have an open/closed (0, ε 2)-rectangle of influence drawing for any ε2 ≥ 0. We also prove that if ε2 > 2 an open/closed (0, ε2)-rectangle of influence drawing of an outerplanar graph can be computed in polynomial area. For values of ε2 such that ε2 ≤ 2, we describe a drawing algorithm that computes (0, ε2)-rectangle of influence drawings of binary trees in area, O(n2+f(ε2)), where f(ε 2) is a logarithmic function that tends to infinity as ε2 tends to zero, and n is the number of vertices of the input tree. © 2013 Springer-Verlag.