Notes on large angle crossing graphs

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2

Exploring the relative importance of crossing number and crossing angle

Recent research has indicated that human graph reading performance can be affected by the size of crossing angle. Crossing angle is closely related to another aesthetic criterion: number of edge crossings. Although crossing number has been previously identified as the most important aesthetic, its relative impact on performance of human graph reading is unknown, compared to crossing angle. In this paper, we present an exploratory user study investigating the relative importance between crossing number and crossing angle. This study also aims to further examine the effects of crossing number and crossing angle not only on task performance measured as response time and accuracy, but also on cognitive load and visualization efficiency. The experimental results reinforce the previous findings of the effects of the two aesthetics on graph comprehension. The study demonstrates that on average these two closely related aesthetics together explain 33% of variance in the four usability measures: time, accuracy, mental effort and visualization efficiency, with about 38% of the explained variance being attributed to the crossing angle. Copyright © 2010 ACM

L-Visibility Drawings of IC-planar Graphs

An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge

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

Bar 1-Visibility Drawings of 1-Planar Graphs

A bar 1-visibility drawing of a graph $G$ is a drawing of $G$ where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an end point of the edge. A graph $G$ is bar 1-visible if $G$ has a bar 1-visibility drawing. A graph $G$ is 1-planar if $G$ has a drawing in a 2-dimensional plane such that an edge crosses at most one other edge. In this paper we give linear-time algorithms to find bar 1-visibility drawings of diagonal grid graphs and maximal outer 1-planar graphs. We also show that recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs are bar 1-visible graphs.Comment: 15 pages, 9 figure