Graphs with large total angular resolution

08/18/2019
by   Oswin Aichholzer, et al.
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The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with n vertices and a total angular resolution greater than 60^∘ is bounded by 2n-6. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least 60^∘ is NP-hard.

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