NDLI: On the Angular Resolution of Planar Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Malitz, Seth Papakostas, Achilleas
Copyright Year	1994
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. 86951, 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, $0 < \alpha < 1$ 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 $\Omega (1/d)$, 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.
Starting Page	172
Ending Page	183
Page Count	12
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480193242931
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	7
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	1994-05-01
Access Restriction	Subscribed
Subject Keyword	max-flow min-cut theorem outerplanar graph planar graph disk-packing angular resolution
Content Type	Text
Resource Type	Article
Subject	Mathematics

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