Related papers: Graphs with large total angular resolution
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…
A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all…
A RAC-drawing of a graph is a straight-line drawing in which every crossing occurs at a right-angle. We show that deciding whether a graph has a RAC-drawing is as hard as the existential theory of the reals, even if we know that every edge…
A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections…
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…
A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced…
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…
We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining…
The Straightness is a measure designed to characterize a pair of vertices in a spatial graph. It is defined as the ratio of the Euclidean distance to the graph distance between these vertices. It is often used as an average, for instance to…
In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way.…
A k-bend right-angle-crossing drawing or (k-bend RAC drawing}, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph…
A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone…
We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.
A forcing set for a perfect matching of a graph is defined as a subset of the edges of that perfect matching such that there exists a unique perfect matching containing it. A complete forcing set for a graph is a subset of its edges, such…
A convex geometric graph is a graph whose vertices are the corners of a convex polygon P in the plane and whose edges are boundary edges and diagonals of the polygon. It is called triangulation-free if its non-boundary edges do not contain…
Given a straight-line drawing of a graph, a segment is a maximal set of edges that form a line segment. Given a planar graph $G$, the segment number of $G$ is the minimum number of segments that can be achieved by any planar straight-line…
There are two particular $\Theta_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Tur\'an number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given $\Theta_6$…
This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are…