Outerplanar Thrackles

We show that a graph drawing is an outerplanar thrackle if and only if, up to an inversion in the plane, it is Reidemeister equivalent to an odd musquash. This establishes Conway’s thrackle conjecture for outerplanar thrackles. We also extend this result in two directions. First, we show that no pair of vertices of an outerplanar thrackle can be joined by an edge in such a way that the resulting graph drawing is a thrackle. Secondly, we introduce the notion of crossing rank; drawings with crossing rank 0 are generalizations of outerplanar drawings. We show that all thrackles of crossing rank 0 are outerplanar. We also introduce the notion of an alternating cycle drawing, and we show that a thrackled cycle is alternating if and only if it is outerplanar.     

Odd Crossing Number and Crossing Number Are Not the Same

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems).     

Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.     

On the Pseudolinear Crossing Number

A drawing of a graph is pseudolinear if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The pseudolinear crossing number of a graph G is the minimum number of pairwise crossings of edges in a pseudolinear drawing of G. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.     

Which Crossing Number Is It Anyway?

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed, twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E₀ be a subset of its edges such that there is a drawing of G, in which every edge belonging to E₀ crosses any other edge an even number of times. Then g can be redrawn so that the elements of E₀ are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-crossing number.     

Genetic algorithms for the 2-page book drawing problem of graphs

The minimisation of edge crossings in a book drawing of a graph is one of the important goals for a linear VLSI design, and the 2-page crossing number of a graph provides an upper bound for the standard planar crossing number. We design genetic algorithms for the 2-page drawings, and test them on the benchmark test suits, Rome graphs and Random Connected Graphs. We also test some circulant graphs, and get better results than previously presented in the literature. Moreover, we formalise three conjectures for certain kinds of circulant graphs,supported by our experimental results.     

A framework and algorithms for circular drawings of graphs

In this paper, we present a framework and two linear time algorithms for obtaining circular drawings of graphs. The first technique produces circular drawings of biconnected graphs and finds a zero crossing circular drawing if one exists. The second technique finds multiple embedding circle drawings. Techniques for the reduction of edge crossings are also discussed. Results of experimental studies are included.     

Linear Arboricity of NIC-Planar Graphs

Acta Mathematicae Applicatae Sinica, English Series - A graph is NIC-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 at...     

Crossing minimization in extended level drawings of graphs

The most popular method of drawing directed graphs is to place vertices on a set of horizontal or concentric levels, known as level drawings. Level drawings are well studied in Graph Drawing due to their strong application for the visualization of hierarchy in graphs. There are two drawing conventions: Horizontal drawings use a set of parallel lines and radial drawings use a set of concentric circles.In level drawings, edges are only allowed between vertices on different levels. However, many real world graphs exhibit hierarchies with edges between vertices on the same level. In this paper, we initiate the new problem of extended level drawings of graphs, which was addressed as one of the open problems in social network visualization, in particular, displaying centrality values of actors. More specifically, we study minimizing the number of edge crossings in extended level drawings of graphs. The main problem can be formulated as the extended one-sided crossing minimization problem between two adjacent levels, as it is folklore with the one-sided crossing minimization problem in horizontal drawings.We first show that the extended one-sided crossing minimization problem is NP-hard for both horizontal and radial drawings, and then present efficient heuristics for minimizing edge crossings in extended level drawings. Our extensive experimental results show that our new methods reduce up to 30% of edge crossings.     

Some provably hard crossing number problems

This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings. While this result has no bearing on the P versus NP question, it is fairly negative with regard to applications. We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines. This work was partially supported by the Center for Telecommunications Research, Columbia University.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Drawing c-planar biconnected clustered graphs

In a graph, a cluster is a set of vertices, and two clusters are said to be non-intersecting if they are disjoint or one of them is contained in the other. A clustered graph C consists of a graph G and a set of non-intersecting clusters. In this paper, we assume that C has a compound planar drawing and each cluster induces a biconnected subgraph. Then we show that such a clustered graph admits a drawing in the plane such that (i) edges are drawn as straight-line segments with no edge crossing and (ii) the boundary of the biconnected subgraph induced by each cluster is a convex polygon.     

On the Minimum Cut of Planarizations

Every drawing of a non-planar graph G in the plane induces a planarization, i.e., a planar graph obtained by replacing edge crossings with dummy vertices. In this paper, we consider the relationship between the capacity of a minimum st-cut in a graph G and its crossing minimal planarizations. We show that these capacities need not be equal. On the other hand, we prove that every such planarization can be efficiently transformed into another crossing minimal planarization that preserves the capacity of a minimum st-cut in G. Furthermore, we extend the result to general (reasonable) planarizations.     

The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log² n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m²/k²) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m³/n²k²), provided that m = Ψ(n²). © 1996 John Wiley & Sons, Inc.     

Two maps on one surface

Two embeddings of a graph in a surface S are said to be “equivalent” if they are identical under an homeomorphism of S that is orientation‐preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be “jointly embedded” if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to find equivalent embeddings of the two graphs that minimize the number of these edge‐crossings; this minimum we call the “joint crossing number” of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re‐embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its “mirror image,” then the joint crossing number can decrease, but not by more than 6.066%. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 198–216, 2001     

A branch and bound algorithm for minimizing the number of crossing arcs in bipartite graphs

Acyclic directed graphs are widely used in many fields of economic and social sciences. This has generated considerable interest in algorithms for drawing “good” maps of acyclic diagraphs. The most important criterion to obtain a readable map of an acyclic graph is that of minimizing the number of crossing arcs. In this paper, we present a branch and bound algorithm for solving the problem of minimizing the number of crossing arcs in a bipartite graph. Computational results are reported on a set of randomly generated test problems.     

An analysis of some linear graph layout heuristics

The worst-case performances of some heuristics for the fixed linear crossing number problem (FLCNP) are analyzed. FLCNP is similar to the 2-page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal “node line” in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In FLCNP, the order of the vertices along the node line is predetermined and fixed. FLCNP belongs to the class of NP-hard optimization problems Masuda et al., 1990. In this paper we show that for each of the heuristics described, there exist classes of n-vertex, m-edge graphs which force it to obtain a number of crossings which is a function of n or m when the optimal number is a small constant. This leaves open the problem of finding a heuristic with a constant error bound for the problem.     

A Framework for Drawing Planar Graphs with Curves and Polylines

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n) × O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) × O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.     

Triple Crossing Numbers of Graphs

We introduce the triple crossing number,a variation of the crossing number,of a graph,which is the minimal number of crossing points in all drawings of the graph with only triple crossings.It is defined to be zero for planar graphs,and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings.In this paper,we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.