The Approximate Rectangle of Influence Drawability Problem

Abstract

Let ε 1,ε 2 be two non negative numbers. An approximate rectangle of influence drawing (also called (ε 1,ε 2)-RID) is a proximity drawing of a graph such that: (i) if u,v are adjacent vertices then their rectangle of influence “scaled down” by the factor \(\frac{1}{1+\varepsilon_{1}}\) does not contain other vertices of the drawing; (ii) if u,v are not adjacent, then their rectangle of influence “blown-up” by the factor 1+ε 2 contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an (ε 1,ε 2)-RID for any ε 1>0 and ε 2>0, while there exist planar graphs which do not admit an (ε 1,0)-RID and planar graphs which do not admit a (0,ε 2)-RID. Then, we investigate (0,ε 2)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0,ε 2)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε 2>2. As for values of ε 2 such that ε 2≤2, we describe a drawing algorithm that computes (0,ε 2)-RIDs of binary trees in area \(O(n^{c + f(\varepsilon_{2})})\), where c is a positive constant, f(ε 2) is a poly-logarithmic function that tends to infinity as ε 2 tends to zero, and n is the number of vertices of the input tree.