On the Angular Resolution of Planar Graphs

It is a well-known fact that every planar graph admits a planar straight-line drawing. The angular resolution of such a drawing is the minimum angle subtended by any pair of incident edges. The angular resolution of the graph is the supremum angular resolution over all planar straight-line drawings of the graph. In a recent paper by Formann et al. [Proc. 31st IEEE Sympos. on Found. of Comput. Sci., 1990, pp. 86–951, 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 ) radians? The present authors show that the answer is yes and that it follows easily from results in the literature on disk-packings. The conclusion is that every planar graph of maximum degree d has angular resolution at least αᵈ radians, 0 < α < 1 constant. In an effort to assess whether this lower bound is existentially tight (up to constant α), a very natural linear program (LP) that bounds the angular resolution of a planar graph the authors analyze from above. The optimal value of this LP is shown to be Ω (1/d), which suggests that the αᵈ lower bound might be improved to Ω ( 1/d ). Although this matter remains unsettled for general planar graphs, Ω ( 1/d ) is shown to be a lower bound on angular resolution for outerplanar graphs. Finally, an infinite family of triangulated planar graphs with maximum degree 6 is constructed such that exponential area is required to draw each member in planar straight-line fashion with angular resolution bounded away from zero.