# On the Red/Blue Spanning Tree Problem

• Sergey Bereg
• Minghui Jiang
• Boting Yang
• Binhai Zhu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

## Abstract

A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is a set of non-crossing straight line segments with endpoints in S. Given a set of red points and a set of blue points in the plane, the red/blue spanning tree problem is to find a geometric spanning tree for red points and a geometric spanning tree for blue points such that the number of crossing points of the two trees is minimum. If no three points are collinear, we show that the minimum number of crossing points is completely determined by the number of maximal red chains on the convex hull of all red points and blue points. We design an optimal algorithm for constructing a geometric spanning tree of all the red points and a geometric spanning tree of all the blue points with the minimum number of crossing points. If collinear points are allowed, we prove that the problem of deciding whether there exists a geometric spanning path of all the red points and a geometric spanning path of all the blue points without crossing is NP-complete.

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### References

1. 1.
Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pp. 617–626 (2002)Google Scholar
2. 2.
Chazelle, B.: On the convex layers of a planar set. IEEE Transactions on Information Theory IT-31, 509–517 (1985)
3. 3.
Fary, I.: On straight line representations of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)
4. 4.
Garey, M., Johnson, D.: Crossing number is NP-complete. SIAM J. Algebraic and Discrete Methods 4, 312–316 (1983)
5. 5.
Hliněný, P.: Crossing number is hard for cubic graphs. Journal of Combinatorial Theory, Series B 96, 455–471 (2006)

## Authors and Affiliations

• Sergey Bereg
• 1
• Minghui Jiang
• 2
• Boting Yang
• 3
• Binhai Zhu
• 4
1. 1.Department of Computer ScienceUniversity of Texas at Dallas
2. 2.Department of Computer ScienceUtah State University
3. 3.Department of Computer ScienceUniversity of Regina
4. 4.Department of Computer ScienceMontana State University