A fully dynamic algorithm for modular decomposition and recognition of cographs

Authors:
Ron Shamir;Roded Sharan
Affiliations:
School of Computer Science, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel;International Computer Science Institute, 1947 Center St., Suite 600, Berkeley, CA
Venue:
Discrete Applied Mathematics - The 1st cologne-twente workshop on graphs and combinatorial optimization (CTW 2001)
Year:
2004

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Abstract

The problem of dynamically recognizing a graph property calls for efficiently deciding if an input graph satisfies the property under repeated modifications to its set of vertices and edges. The input to the problem consists of a series of modifications to be performed on the graph. The objective is to maintain a representation of the graph as long as the property holds, and to detect when it ceases to hold. In this paper, we solve the dynamic recognition problem for the class of cographs and some of its subclasses. Our approach is based on maintaining the modular decomposition tree of the dynamic graph, and using this tree for the recognition. We give the first fully dynamic algorithm for maintaining the modular decomposition tree of a cograph. We thereby obtain fully dynamic algorithms for the recognition of cographs, threshold graphs, and trivially perfect graphs. All these algorithms work in constant time per edge modification and O(d) time per d-degree vertex modification.