Maximizing Maximal Angles for Plane Straight-Line Graphs

Abstract

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

“…A somewhat simpler construction was subsequently proposed by Ackerman et al [1]. Aichholzer et al [2] have also obtained this result (together with additional related results), independently. However, in all these papers, the goal is to construct an α-ST (for α = π/3) and not an α-MST.The problem of computing an α-MST is similar in flavor to the problem of computing a Euclidean minimum weight degree-k spanning tree, which has been studied extensively (see, e.g., [4,10,15,16,18]).…”

“…A somewhat simpler construction was subsequently proposed by Ackerman et al [1]. Aichholzer et al [2] have also obtained this result (together with additional related results), independently. However, in all these papers, the goal is to construct an α-ST (for α = π/3) and not an α-MST.…”

“…An α-MST of P is then an α-ST of P of minimum weight. For α < π/3, an α-ST does not always exist, and, for α ≥ π/3, it always exists [1,2,9]. In this paper, we study the problem of computing an α-MST for several common values of α.Motivated by wireless networks, we formulate the problem in terms of directional antennas.…”

Drawing Hamiltonian Cycles with no Large Angles

“…Related problems and results. Various angle conditions imposed on geometric graphs (graphs with straight-line edges) drawn on a given set of points have been studied in [3,4,5,6,12]. In particular, Fekete and Woeginger [12] have focused on rotation angles of Hamiltonian cycles and paths and raised many challenging questions.…”

“…Aichholzer et al [3] studied similar questions for planar geometric graphs. Among other results, they showed that any point set in general position in the plane admits a non-intersecting Hamiltonian (spanning) path with the property that each rotation angle is at most 3π/4.…”