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Báo cáo toán học: "Characterization of [1, k]-Bar Visibility Trees"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Characterization of [1, k]-Bar Visibility Trees. | Characterization of 1 k -Bar Visibility Trees Guantao Chen 1 Joan P. Hutchinson2 Ken Keating3 Jian Shen4 1 Georgia State University Atlanta GA 30303 gchen@gsu.edu 2 Macalester College St. Paul MN 55105 hutchinson@macalester.edu 3 Emory University Atlanta GA 30322 kekeati@emory.edu 4 Texas State University San Marcos TX 78666 js48@txstate.edu Submitted May 27 2006 Accepted Oct 19 2006 Published Oct 27 2006 Mathematics Subject Classification 05E25 Abstract A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed non-degenerate vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to 1 k -bar-visibility graphs by allowing the lengths of the bars to be between 1 k and 1. We completely characterize these graphs for trees. We establish an algorithm with complexity O kn to determine whether a tree with n vertices has a 1 k -bar-visibility representation. In the course of developing the algorithm we study a special case of the knapsack problem Partitioning a set of positive integers into two sets with sums as equal as possible. We give a necessary and sufficient condition for the existence of such a partition. 1 Introduction A bar-visibility graph or BVG for short is a graph whose vertices can be represented in the plane by disjoint horizontal bars such that two vertices are adjacent if and only if there Partially supported by NSA grant H98230-04-1-0300 and NSF grant DMS-0500951 THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R90 1 is an unobstructed non-degenerate vertical band of visibility between the corresponding bars. The study of BVGs is motivated by VLSI design. Several layout compaction strategies for VLSI are based on the concept of visibility between parallel segments. Two parallel segments are visible if they can be joined by a segment orthogonal to them without .