A Dualizable Representation for Imbedded Graphs

A new representation is described for graphs imbedded in pseudomanifoldr. The graphs may have loops and parallel edges, and they may be disconnected. It is shown that our representation has a well-dejined dual that naturally extends the definition of surface duality. This representation is easily transformed to a data structure with a set of operators for modifying a graph, a set of query operators, and a set of navigation operators. Three variants of the data structure are described. For a graph with n vertices, the first provides O(1)-time updates and O(n)-time queries. The second provides O(n)-time updates and O(1)-time queries. The third, a rradeoff between the first two, provides O(1og n)-time updates and queries. The space used in all variants is linear in the size of the graph.