New results on drawing angle graphs

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Abstract

An angle graph is a graph with a fixed cyclic order of the edges around each vertex and an angle specified for every pair of consecutive edges incident on a vertex. We study the problem of constructing a drawing of an angle graph that preserves its angles, and present several new results.

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    • We disprove the conjectures of Vijayan (1986) about the relationship between the planarity of an angle graph and the planarity of its biconnected components.

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    • We show that testing an angle graph for planarity is NP-hard.

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    • Using our NP-hardness result, we show that given a triconnected planar graph and an angle α, the problem of determining whether it admits a planar straight-line drawing in which the angle between any two consecutive edges incident on the same vertex is at least α, is NP-hard.

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    • A multilayered angle graph is one whose edges are assigned to a given set of layers. A multilayered angle graph is multiplanar if it admits a drawing in which edges assigned to the same layer do not cross. We prove that testing a multilayered angle graph for multiplanarity is NP-hard even if we restrict the angles to be multiples of 90° and the number of layers to two.

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    • We give a simple linear time algorithm for testing whether a series-parallel angle graph admits a (nonplanar) drawing that preserves its angles.

Keywords

Graph drawing
Layout
Angle
Constraints
Resolution area
Series-parallel graphs

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Part of research was done while the author was at the Center for Geometric Computing, Department of Computer Science, Brown University, USA. Research supported in part by the National Science Foundation under grants CCR-9007851 and CCR-9423847, by the U.S. Army Research Office under grants DAAL03-91-G-0035, DAAH04-93-0134, and 34990-MA-MUR, and by the Office of Naval Research and the Advanced Research Projects Agency under contract N00014-91-J-4052, ARPA order 8225.