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On Plane Constrained Bounded-Degree Spanners

Abstract

Let P be a finite set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted \(\mathord { Vis}(P,S)\), has vertex set P and an edge for each pair of vertices uv in P for which no line segment of S properly intersects uv. We show that the constrained half-\(\theta _6\)-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of \(\mathord { Vis}(P,S)\). We then show how to construct a plane 6-spanner of \(\mathord { Vis}(P,S)\) with maximum degree \(6+c\), where c is the maximum number of segments of S incident to a vertex.

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Acknowledgements

This work is supported in part by the Natural Science and Engineering Research Council of Canada, Carleton University’s President’s 2010 Doctoral Fellowship, the Ontario Ministry of Research and Innovation, and the Danish Council for Independent Research, Natural Sciences, Grant DFF-1323-00247, and JST ERATO Grant No. JPMJER1201, Japan.

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Correspondence to André van Renssen.

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An extended abstract of this paper appeared in the proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012) [5].

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Bose, P., Fagerberg, R., van Renssen, A. et al. On Plane Constrained Bounded-Degree Spanners. Algorithmica 81, 1392–1415 (2019). https://doi.org/10.1007/s00453-018-0476-8

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Keywords

  • Plane spanners
  • Bounded-degree
  • Constraints
  • Visibility graph