相关论文: On the rectilinear crossing number of complete bal…

A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of edges crosses precisely once. A special case are…

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in…

The crossing number of a graph $G$ is the minimum number of crossings in a drawing of $G$ in the plane. A rectilinear drawing of a graph $G$ represents vertices of $G$ by a set of points in the plane and represents each edge of $G$ by a…

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing…

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 $G$,…

The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one point (counting endpoints), no three…

It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the…

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…

Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show…

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\Delta n)$, where $G$…

In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the $d$-dimensional rectilinear drawing of a…

We introduce the \textit{crossing profile} of a drawing of a graph. This is a sequence of integers whose $(k+1)^{\text{th}}$ entry counts the number of edges in the drawing which are involved in exactly $k$ crossings. The first and second…

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…

In this work, we study a mathematically rigorous metric of a graph visualization quality under conditions that relate to visualizing a bipartite graph. Namely we study rectilinear crossing number in a special arrangement of the complete…

A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G…

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant…

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of…

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of…

We introduce the triple crossing number, a variation of crossing number, of a graph, which is the minimal number of crossing points in all drawings with only triple crossings of the graph. It is defined to be zero for a planar graph, and to…

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. For such drawings to be readable, special care is needed in order to keep the number…