Counting spanning trees using modular decomposition

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)(q,q−4)-graph for fixed q  , a P4P4-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.