Dynamically switching vertices in planar graphs

Abstract

We consider graphs whose vertices may be in one of two different states: either on or off. We wish to maintain dynamically such graphs under an intermixed sequence of updates and queries. An update may reverse the status of a vertex, by switching it either on or off, and may insert a new edge or delete an existing edge. A query tests properties of the subgraph induced by the vertices that are on. We give efficient algorithms that maintain information about connectivity on planar graphs in O(log3 n) amortized time per query, insert, delete, switch-on and switchoff operation over sequences of at least Ω(n) operations, where n is the number of vertices of the graph.