Related papers: Crossing Numbers and Stress of Random Graphs
The stress of a vertex in a graph is the number of geodesics passing through it (A. Shimbel, 1953). A graph is $k$-stress regular if stress of each of its vertices is $k$. In this paper, we investigate some results and compute stress of…
Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is…
A straight-line drawing of a graph $G$ is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph $G$,…
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…
We propose a random bipartite graph with weights assigned to both parts of the vertex sets. Edges are formed independently with probabilities that depend on these weights. This bipartite graph naturally gives rise to a random intersection…
A graph is crossing-critical if the removal of any of its edges decreases its crossing number. This work is motivated by the following question: to what extent is crossing- criticality a property that is inherent to the structure of a…
We consider the number of crossings in a random labelled tree with vertices in convex position. We give a new proof of the fact that this quantity is asymptotically Gaussian with mean $n^2/6$ and variance $n^3/45$. Furthermore, we give an…
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two…
We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized…
We offer a solution to a long-standing problem in the physics of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity -- the propensity for two neighbors of a network node also to be…
A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph $G$ is the minimum number of pairwise…
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…
We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.
We consider the minimization of edge-crossings in geometric drawings of graphs $G=(V, E)$, i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Bienstock, '91]. In contrast to theory…
In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically…
A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…
In this paper we deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and…
Recent empirical research has indicated that human graph reading performance improves when crossing angles increase. However, crossing angle has not been used as an aesthetic criterion for graph drawing algorithms so far. In this paper, we…
A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…