On plane spanning trees and cycles of multicolored point sets with few intersections

Authors:
M. Kano;C. Merino;J. Urrutia
Affiliations:
Department of Computer and Information Science, Ibaraki University, Hitachi 316-8511, Japan;Instituto de Matemáticas, U.N.A.M. Área de la investigación científica, Circuito Exterior, Ciudad Universitaria Coyoacán 04510, México D.F., Mexico;Instituto de Matemáticas, U.N.A.M. Área de la investigación científica, Circuito Exterior, Ciudad Universitaria Coyoacán 04510, México D.F., Mexico
Venue:
Information Processing Letters
Year:
2005

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Abstract

Let P1, ..., Pk be a collection of disjoint point sets in R2 in general position. We prove that for each 1 ≤ i ≤ k we can find a plane spanning tree Ti of Pi such that the edges of T1 ,..., Tk intersect at most (k - 1)(n - k) + ½k(k - 1), where n is the number of points in P1 ∪ ... ∪ Pk. If the intersection of the convex hulls of P1,...., Pk is nonempty, we can find k spanning cycles such that their edges intersect at most (k - 1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8n times, where |P ∪ Q| = n (the weight of an edge is its length).