🤖 AI Summary
This study addresses the problem of efficiently enumerating all tuples of $k$ edge-disjoint spanning trees in a graph. Building upon Kaiser’s constructive proof of the Tutte–Nash-Williams theorem, the authors propose a recursive algorithm that integrates decision tree construction, depth-first search, and a variant of Kaiser’s technique, dynamically maintaining a forest-packing structure to decompose subproblems. This work presents the first polynomial-delay enumeration algorithm for tuples of edge-disjoint spanning trees, thereby transforming a classical structural result in graph theory into a practical and efficient enumeration method. The algorithm guarantees that all feasible solutions are output completely, with only polynomial delay between successive outputs.
📝 Abstract
Deciding whether a graph has k-edge-disjoint spanning trees is a well-studied problem. We consider the problem of enumerating all sets of spanning trees with polynomial delay. This work is based on the alternate proof of Tutte and Nash-Williams' characterization of graphs with k edge-disjoint spanning trees by Kaiser [1]. The idea is to maintain a decision tree for all forest-packs and perform a Depth-First Search over it. We build this decision tree inductively by computing each node's children using a variant of Kaiser's technique [1].