On the angular resolution of planar graphs

Abstract

A famous theorem of I. Fàry states that any planar graph can be drawn in the plane so that all edges are straight-line segments and no two edges cross. The angular resolution of such a drawing is the minimum angle subtended by any pair of incident edges. The angular resolution of a planar graph is the maximum angular resolution over all such planar straight-line drawings of the graph. In a recent paper by Formann et al., Drawing graphs in the plane with high resolution, Symp. on Found, of Comp. Sci. (1990), the following question is posed: does there exist a constant r(d) > 0 such that every planar graph of maximum degree d has angular resolution ≥ r(d)? We answer this question in the affirmative by showing that any planar graph of maximum degree d has angular resolution at least αd radians where 0 < α < 1 is a constant. In an effort to assess whether or not this lower bound is existentially tight (up to constant a), we analyze a very natural linear program that bounds the angular resolution of any fixed planar graph G from above. The optimal value of this LP is shown to be ω(l/d). This suggests that our ad lower bound might be improved to ω(l/d), although currently, we are unable to settle this issue for general planar graphs. For the class of outerplanar graphs with triangulated interior and maximum degree d, we show not only that ω(l/d) is lower bound on angular resolution, but in fact, this angular resolution can be achieved in a planar straight-line drawing where all interior faces are similar isosceles triangles. Additional results are contained in the full paper.