Revisiting the Top-Down Computation of BDD of Spanning Trees of a Graph and Its Tutte Polynomial

Abstract

Revisiting the Sekine-Imai-Tani top-down algorithm to compute the BDD of all spanning trees and the Tutte polynomial of a given graph, we explicitly analyze the Fixed-Parameter Tractable (FPT) time complexity with respect to its (proper) pathwidth, pw (ppw), and obtain a bound of O^*(Bell_{min{pw}+1,ppw}}), where Bell_n denotes the n-th Bell number, defined as the number of partitions of a set of n elements. We further investigate the case of complete graphs in terms of Bell numbers and related combinatorics, obtaining a time complexity bound of Bell_{n-O(n/log n)}.