Graphs with large total angular resolution

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with $n$ vertices and a total angular resolution greater than $60^{\circ}$ is bounded by $2n-6$. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least $60^{\circ}$ is NP-hard.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

Maximizing the Total Resolution of Graphs

A major factor affecting the readability of a graph drawing is its resolution. In the graph drawing literature, the resolution of a drawing is either measured based on the angles formed by consecutive edges incident to a common node (angular resolution) or by the angles formed at edge crossings (crossing resolution). In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution. To the best of our knowledge, this is the first time where the problem of maximizing the total resolution of a drawing is studied. The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings). In addition, we present and experimentally evaluate a force-directed based algorithm that constructs drawings of large total resolution

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Optimal 3D Angular Resolution for Low-Degree Graphs

We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5-degree angles, i.e., the angular resolution of the diamond lattice, between any two edge segments meeting at a vertex or bend.Comment: 18 pages, 10 figures. Extended version of paper to appear in Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 201

Experimental analysis of the accessibility of drawings with few segments

The visual complexity of a graph drawing is defined as the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges, e.g., one needs only one line segment to draw two collinear incident edges. We study the question if drawings with few segments have a better aesthetic appeal and help the user to asses the underlying graph. We design an experiment that investigates two different graph types (trees and sparse graphs), three different layout algorithms for trees, and two different layout algorithms for sparse graphs. We asked the users to give an aesthetic ranking on the layouts and to perform a furthest-pair or shortest-path task on the drawings.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Trees with Convex Faces and Optimal Angles

We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.Comment: 12 pages, 10 figures. To appear at 14th Int. Symp. Graph Drawing, 200

Double radiative pion capture on hydrogen and deuterium and the nucleon's pion cloud

We report measurements of double radiative capture in pionic hydrogen and pionic deuterium. The measurements were performed with the RMC spectrometer at the TRIUMF cyclotron by recording photon pairs from pion stops in liquid hydrogen and deuterium targets. We obtained absolute branching ratios of $(3.02 \pm 0.27 (stat.) \pm 0.31 (syst.)) \times 10^{-5}$ for hydrogen and $(1.42 \pm ^{0.09}_{0.12} (stat.) \pm 0.11 (syst.)) \times 10^{-5}$ for deuterium, and relative branching ratios of double radiative capture to single radiative capture of $(7.68 \pm 0.69(stat.) \pm 0.79(syst.)) \times 10^{-5}$ for hydrogen and $(5.44 \pm^{0.34}_{0.46}(stat.) \pm 0.42(syst.)) \times 10^{-5}$ for deuterium. For hydrogen, the measured branching ratio and photon energy-angle distributions are in fair agreement with a reaction mechanism involving the annihilation of the incident $\pi^-$ on the $\pi^+$ cloud of the target proton. For deuterium, the measured branching ratio and energy-angle distributions are qualitatively consistent with simple arguments for the expected role of the spectator neutron. A comparison between our hydrogen and deuterium data and earlier beryllium and carbon data reveals substantial changes in the relative branching ratios and the energy-angle distributions and is in agreement with the expected evolution of the reaction dynamics from an annihilation process in S-state capture to a bremsstrahlung process in P-state capture. Lastly, we comment on the relevance of the double radiative process to the investigation of the charged pion polarizability and the in-medium pion field.Comment: 44 pages, 7 tables, 13 figures, submitted to Phys. Rev.