Counting spanning trees in graphs using modular decomposition

Authors:
Stavros D. Nikolopoulos;Leonidas Palios;Charis Papadopoulos
Affiliations:
Department of Computer Science, University of Ioannina, Greece;Department of Computer Science, University of Ioannina, Greece;Department of Mathematics, University of Ioannina, Greece
Venue:
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
Year:
2011

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Abstract

In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q, q - 4)-graph for fixed q, a P4-tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in time linear in the size of the graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first linear-time algorithm for the counting problem in the considered graph classes.