Crossing minimisation Heuristics for 2-page drawings

The minimisation of edge crossings in a book drawing of a graph G is one of important goals for a linear VLSI design, and the two-page crossing number of a graph G provides an upper bound for the standard planar crossing number. We propose several new heuristics for the 2-page drawing problem and test them on benchmark test sets like Rome graphs, Random Connected Graphs and some typical graphs. We get exact results of some structural graphs, and compare some of the experimental results with the one in paper

New circular drawing algorithms

In the circular (other alternate concepts are outerplanar, convex and one-page) drawing one places vertices of a n-vertex m-edge connected graph G along a circle, and the edges are drawn as straight lines. The smallest possible number of crossings in such a drawing of the graph G is called circular (outerplanar, convex, or one-page) crossing number of the graph G. This paper addresses heuristic algorithms to find an ordering of vertices to minimise the number of crossings in the corresponding circular drawing of the graph. New algorithms to find low crossing circular drawings are presented, and compared with algorithm of Makinen, CIRCULAR+ algorithm of Six and Tollis and algorithm of Baur and Brandes. We get better or comparable results to the other algorithms

An improved neural network model for the two-page crossing number problem

The simplest graph drawing method is that of putting the vertices of a graph on a line and drawing the edges as half-circles either above or below the line. Such drawings are called 2-page book drawings. The smallest number of crossings over all 2-page drawings of a graph G is called the 2-page crossing number of G. Cimikowski and Shope have solved the 2-page crossing number problem for an n-vertex and m-edge graph by using a Hopfield network with 2m neurons. We present here an improved Hopfield modelwith m neurons. The new model achieves much better performance in the quality of solutions and is more efficient than the model of Cimikowski and Shope for all graphs tested. The parallel time complexity of the algorithm, without considering the crossing number calculations, is O(m), for the new Hopfield model with m processors clearly outperforming the previous algorithm

Experimental Evaluation of Book Drawing Algorithms

A $k$-page book drawing of a graph $G=(V,E)$ consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the $k$ pages, which are half-planes bounded by the spine. In a book drawing, two edges cross if and only if they are assigned to the same page and their vertices alternate along the spine. Crossing minimization in a $k$-page book drawing is NP-hard, yet book drawings have multiple applications in visualization and beyond. Therefore several heuristic book drawing algorithms exist, but there is no broader comparative study on their relative performance. In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page