Related papers: Algorithms and Bounds for Drawing Non-planar Graph…
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 $G$,…
A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small…
It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further…
We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
A fan is a set of edges with a single common endpoint. A graph is fan-crossing if it admits a drawing in the plane so that each edge is crossed by edges of a fan. It is fan-planar if, in addition, the common endpoint is on the same side of…
Given a graph $G$, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of $G$ with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but---to the…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
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 multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges…
We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest…
The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge…
We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where…
The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back…
When drawing graphs whose edges and nodes contain text or graphics, such informa tion needs to be displayed without overlaps, either as part of the initial layout or as a post-processing step. The core problem in removing overlaps lies in…
A crossing-free morph is a continuous deformation between two graph drawings that preserves straight-line pairwise noncrossing edges. Motivated by applications in 3D morphing problems, we initiate the study of morphing graph drawings in the…
This paper studies questions about duality between crossings and non-crossings in graph drawings via the notions of thickness and antithickness. The "thickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the…
We initiate the systematic study of the following Tur\'an-type question. Suppose $\Gamma$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$…
The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…