1994/05/01 by Seth M. Malitz, Achilleas Papakostas · 3 citations
Computer Science · Environmental Science · Mathematics · #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #Remote Sensing and LiDAR Applications #Mathematics #Combinatorics #Planar graph #Planar #Omega #Degree (music) #Graph #Upper and lower bounds #Infimum and supremum #Angular resolution (graph drawing) #Resolution (logic) #Discrete mathematics #Geometry #Mathematical analysis #Physics #Computer science #Quantum mechanics
paper · doi:10.1137/s0895480193242931
openalex publication_date 1994/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
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.