Constrained Tri-Connected Planar Straight Line Graphs
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.
We are grateful to the anonymous referee who pointed out several errors and omissions in an earlier version of the proof of Lemma 3.1.
This material is based upon work supported by the National Science Foundation under Grant No. 0830734. Research by C. D. Tóth was also supported by NSERC grant RGPIN 35586. Preliminary results have been presented at the 26th European Workshop on Computational Geometry (2010, Dortmund) and at the 20th Annual Fall Workshop on Computational Geometry (2010, Stony Brook, NY).
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