A completely dynamic algorithm for split graphs

We present a fully dynamic algorithm for split graphs that supports the following types of operations: (1) query whether deleting or inserting an edge preserves the split property, (2) query whether inserting a new vertex with a given neighborhood in the current graph preserves the split property, (3) insert or delete an edge or a vertex when the split property is preserved, (4) insert an edge or a vertex even when the split property is not preserved, adding a minimal or a minimum set of additional edges to keep the graph split, (5) delete an edge even when the split property is not preserved, deleting a minimal or a minimum set of edges from the current graph to keep the graph split. Usually, fully dynamic algorithms for recognizing and maintaining a class of graphs support operations of type (1), (2), and (3). Because of the additional operations we call our algorithm completely dynamic. As an interesting consequence, we show that the above operations very easily lead to the following results, in addition to the dynamic algorithm: a linear-time vertex incremental certifying algorithm for recognizing split graphs, and a linear-time vertex incremental algorithm for computing a minimal split completion of an arbitrary graph. Both of these algorithms match the best known algorithms for these purposes, but have the advantage og being vertex incremental thusly allowing the input to be given on-line. Finally, by the dynamic operations of type (4) and (5), we show that the following problems can be solved in linear time: adding a minimum number of edges to a given split+1v or split+1e graph to obtain a split graph, and removing a minimum number of edges from a given split−1e graph to obtain a split graph.