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To deal with the planarization problem widely used in many applications including routing very-large-scale integration (VLSI) circuits, this paper points out that only when its vertices are arranged in some specific order in a line can a planar graph be embedded on a line without any cross connections or cross edges. Energy function is proposed to meet the need of embedding a graph on a single line and route it correctly. A Hopfield network is designed according to the proposed energy func-tion for such embedding and routing. The advantage of the proposed method is that it not only can detect if a graph is a planar one or not, but also can embed a planar graph or the maximal planar subgraph of a non-planar graph on a single line. In addition, simulated annealing is employed for helping the network to escape from local minima during the running of the Hopfield network. Experiments of the proposed method and its comparison with some existent conventional methods were performed and the results indicate that the proposed method is of great fea-sibility and effectiveness especially for the planarization problem of large graphs.
To deal with the planarization problem widely used in many applications including routing very-large-scale integration (VLSI) circuits, this paper points out that only when its vertices are arranged in some specific order in a line can a planar graph be embedded on a line without any cross connections or cross edges. Energy function is proposed to meet the need of embedding a graph on a single line and route it correctly. A Hopfield network is designed according to the proposed energy func- tion for such embedding and routing. advantage of the proposed method is that it not only can detect if a graph is a planar one or not, but also can embed a planar graph or the maximal planar subgraph of a non-planar graph on a single line. is employed for helping the network to escape from local minima during the running of the Hopfield network. Experiments of the proposed method and its comparison with some existent conventional methods were performed and the re sults indicate that the proposed method is of great fea-sibility and effectiveness especially for the planarization problem of large graphs.