Counting spanning trees using modular decomposition

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 P 4 -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.

The number and degree distribution of spanning trees in the Tower of Hanoi graph

“…The problem of finding the number of spanning trees of a finite graph is a relevant and long standing question. It has been considered in different areas of mathematics [1], physics [2], and computer science [3], since its introduction by Kirchhoff in 1847 [4]. This graph invariant is a parameter that characterizes the reliability of a network [5,6,7] and is related to its optimal synchronization [8] and the study of random walks [9].…”