Fully dynamic algorithms for chordal graphs and split graphs

  • Authors:

  • Louis Ibarra

  • Affiliations:

  • DePaul University, Chicago, IL

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2008

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Abstract

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 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(n log n) time. We also present a data structure that allows a deletion query to run in O(&sqrt;m) 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, and (2) delete or insert an arbitrary edge.