We prove that all planar graphs have an open/closed (ε_1, ε_2)-rectangle of influence drawing for ε_1＞ 0 and ε_2 ＞ 0, while there are planar graphs which do not admit an open/closed (ε_1,0)-rectangle of influence drawing and planar graphs which do not admit a (0,ε_2)-rectangle of influence drawing. We then show that all outerplanar graphs have an open/closed (0, ε_2)-rectangle of influence drawing for any ε_2≥ 0. We also prove that if ε_2 ＞ 2 an open/closed (0,ε_2)-rectangle of influence drawing of an outerplanar graph can be computed in polynomial area. For values of ε_2 such that ε_2 ≤2, we describe a drawing algorithm that computes (0, ε_2)-rectangle of influence drawings of binary trees in area O(n~(2+f(ε_2))), 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.
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