Related papers: One-Bend Drawings of Outerplanar Graphs Inside Sim…
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…
Point-set embeddings and large-angle crossings are two areas of graph drawing that independently have received a lot of attention in the past few years. In this paper, we consider problems in the intersection of these two areas. Given the…
A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…
We prove that for every graph $G$, given fixed locations for the vertices of $G$ in $\mathbb{Z}^3$, there is a three-dimensional grid-drawing of $G$ with one bend per edge. The best previous bound was three bends per edge.
Given a planar graph $G$ and an integer $b$, OrthogonalPlanarity is the problem of deciding whether $G$ admits an orthogonal drawing with at most $b$ bends in total. We show that OrthogonalPlanarity can be solved in polynomial time if $G$…
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…
We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge…
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any…
We consider the problem of computing 2-center in maximal outerplanar graph. In this problem, we want to find an optimal solution where two centers cover all the vertices with the smallest radius. We provide the following result. We can…
A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times…
We describe a set of $\Delta -1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $\Delta \geq 4$; this establishes a new upper bound of $\Delta-1$ on the 1-bend planar slope number. By universal we…
A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most $k$ in a geometric graph $G$ is self-intersecting we call $G$ $k$-locally plane. The main result of this paper is a construction…
A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite $1$-planar graphs with prescribed numbers of vertices in partite sets. Bipartite…
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of…
A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…
We present space-efficient algorithms for computing cut vertices in a given graph with $n$ vertices and $m$ edges in linear time using $O(n+\min\{m,n\log \log n\})$ bits. With the same time and using $O(n+m)$ bits, we can compute the…
A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number $\chi'_l(G)$ of any outer-1-planar graph $G$ with…
Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial…
We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…
We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using…