A Hopfield neural network model for the outerplanar crossing number problem

In the outerplanar (other alternate concepts are circular or 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 minimal number of crossings over all outerplanar drawings of the graph G is called the outerplanar (circular, convex, or one-page) crossing number of the graph G. To find a drawing achieving the minimum crossing number is an NP-hard problem. In this work we investigate the outerplanar crossing number problem with a Hopfield neural network model, and improve the convergence of the network by using the Hill Climbing algorithm with local movement. We use two kinds of energy functions, and compare their convergence. We also test a special kind of graphs, complete p-partite graphs. The experimental results show the neural network model can achieve crossing numbers close to the optimal values of the graphs tested.