Dual Eulerian properties of plane multigraphs

A plane multigraph is said to be dual Eulerian if both it and its dual contain an Euler path or circuit and the Euler paths have corresponding edge sequences. In this paper several properties of plane multigraphs are derived, and a necessary and sufficient condition for a plane multigraph to be dual Eulerian is given. Although the necessary and sufficient condition for a multigraph to be Eulerian is somewhat trivial, the necessary and sufficient condition for a plane multigraph to be dual Eulerian is not. Nevertheless, the question of whether or not a plane multigraph is dual Eulerian can be answered in time proportional to a linear function of the number of edges of the graph, and an algorithm that answers this question is presented in this paper. This theory can be applied to the layout synthesis of functional cells for Complementary Metal-Oxide Semiconductor Very Large-Scale Integrated circuits.