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The Complexity of Drawing a Graph in a Polygonal Region

  • Anna LubiwEmail author
  • Tillmann Miltzow
  • Debajyoti Mondal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

Abstract

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. A special case is the problem of extending a partial planar graph drawing, which was proved NP-hard by Patrignani. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open for a simply connected region.

We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region.

Notes

Acknowledgment

We would like to thank Günter Rote, who discussed with the second author the turn gadget in the context of the \(\textsc {Art~Gallery~Problem}\).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Anna Lubiw
    • 1
    Email author
  • Tillmann Miltzow
    • 2
  • Debajyoti Mondal
    • 3
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Université libre de Bruxelles (ULB)BrusselsBelgium
  3. 3.Department of Computer ScienceUniversity of SaskatchewanSaskatoonCanada

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