Fully dynamic algorithms for chordal graphs and split graphs

We present the first dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality; and (2) delete or insert an arbitrary edge, provided it preserves chordality. We give two implementations. In the first, each operation runs in <i>O</i>(<i>n</i>) time, where <i>n</i> is the number of vertices. In the second, an insertion query runs in <i>O</i>(log² <i>n</i>) time, an insertion in <i>O</i>(<i>n</i>) time, a deletion query in <i>O</i>(<i>n</i>) time, and a deletion in <i>O</i>(<i>n</i> log <i>n</i>) time. We also present a data structure that allows a deletion query to run in <i>O</i>(&sqrt;m) time in either implementation, where <i>m</i> is the current number of edges. Updating this data structure after a deletion or insertion requires <i>O</i>(<i>m</i>) time. We also present a very simple dynamic algorithm that supports each of the following operations in <i>O</i>(1) time on a general graph: (1) query whether the graph is split, and (2) delete or insert an arbitrary edge.