Every Set of Disjoint Line Segments Admits a Binary
نویسندگان
چکیده
Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist conngurations of line segments where any such tree requires degree 3. We provide an O(n log n) time algorithm for constructing such a tree, and show that this is optimal.
منابع مشابه
Pointed Encompassing Trees
It is shown that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree, that is, a spanning tree on the segment endpoints that contains all input segments, has maximal degree three, and such that every vertex is incident to an angle greater than π. As a consequence, it follows that every set of disjoint line segments has a bounded degree pseudo-trian...
متن کاملPointed Binary Encompassing Trees
We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T , that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v ∈ T is pointed, that is, v has an incident angle greater than π. Such a tree can be completed to a minimum pseudo-triangulation. In particular, it follo...
متن کاملThe (non-)existence of perfect codes in Lucas cubes
A Fibonacci string of length $n$ is a binary string $b = b_1b_2ldots b_n$ in which for every $1 leq i < n$, $b_icdot b_{i+1} = 0$. In other words, a Fibonacci string is a binary string without 11 as a substring. Similarly, a Lucas string is a Fibonacci string $b_1b_2ldots b_n$ that $b_1cdot b_n = 0$. For a natural number $ngeq1$, a Fibonacci cube of dimension $n$ is denoted by $Gamma_n$ and i...
متن کاملAlternating paths through disjoint line segments
We show that every segment endpoint visibility graph on disjoint line segments in the plane admits an alternating path of length , answering a question of Bose. This bound is optimal apart from a constant factor. We also give bounds on the constants hidden by the asymptotic notation.
متن کاملTrees in simple Polygons
We prove that every simple polygon contains a degree 3 tree encompassing a prescribed set of vertices. We give tight bounds on the minimal number of degree 3 vertices. We apply this result to reprove a result from Bose et al. [3] that every set of disjoint line segments in the plane admits a binary tree. Introduction Recently many papers have been published regarding the augmentation of discret...
متن کامل