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Constrained Tri-Connected Planar Straight Line Graphs

In: Thirty Essays on Geometric Graph Theory

Author

Listed:
  • Marwan Al-Jubeh

    (Tufts University, Department of Computer Science)

  • Gill Barequet

    (Tufts University, Department of Computer Science
    Technion, Department of Computer Science)

  • Mashhood Ishaque

    (Tufts University, Department of Computer Science)

  • Diane L. Souvaine

    (Tufts University, Department of Computer Science)

  • Csaba D. Tóth

    (Tufts University, Department of Computer Science
    University of Calgary, Department of Mathematics and Statistics)

  • Andrew Winslow

    (Tufts University, Department of Computer Science)

Abstract

It is known that for any set V of n ≥ 4 points in the plane, not in convex position, there is a 3-connected planar straight line graph G = (V, E) with at most 2n − 2 edges, and this bound is the best possible. We show that the upper bound | E | ≤ 2n continues to hold if G = (V, E) is constrained to contain a given graph G 0 = (V, E 0), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i.e., a collection of simple polygons), but no edge in E 0 is a proper diagonal of the convex hull of V. Since there are 1- and 2-factors with n vertices for which any 3-connected augmentation has at least 2n − 2 edges, our bound is nearly tight in these cases. We also examine possible conditions under which this bound may be improved, such as when G 0 is a collection of interior-disjoint convex polygons in a triangular container.

Suggested Citation

  • Marwan Al-Jubeh & Gill Barequet & Mashhood Ishaque & Diane L. Souvaine & Csaba D. Tóth & Andrew Winslow, 2013. "Constrained Tri-Connected Planar Straight Line Graphs," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 49-70, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_5
    DOI: 10.1007/978-1-4614-0110-0_5
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