Fully dynamic algorithms for chordal graphs and split graphs

We present the rst 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, (2) delete or insert an arbitrary edge, provided it preserves chordality. We give two implementations. In the rst, each operation runs in O(n) time, where n is the number of vertices. In the second, an insertion query runs in O(log2 n) time, an insertion in O(n) time, a deletion query in O(n) time, and a deletion in O(nlogn) time. We also present a data structure that allows a deletion query to run in O(pm) time in either implementation, where m is the current number of edges. Updating this data structure after a deletion or insertion requires O(m) time. We also present a very simple dynamic algorithm that supports each of the following operations in O(1) time on a general graph: (1) query whether the graph is split, (2) delete or insert an arbitrary edge.

• Type: Journal
• Year: 2008
• Where: TALG
• Authors: Louis Ibarra
• Keywords: Arbitrary Edge, Clique Tree Representation, Rst Dynamic Algorithm, TALG 2008