On the Angular Resolution of Planar Graphs

Abstract

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. 86-95], 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 $\geq r(d)$ 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 $\alpha^d$ radians, $0 < \alpha < 1$ a constant. In an effort to assess whether this lower bound is existentially tight (up to constant $\alpha$), 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 $\Omega(a/d)$, which suggests that the $\alpha^d$ lower bound might be improved to $\Omega(1/d)$. Although this matter remains unsettled for general planar graphs, $\Omega(1/d)$ 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.