Reconfiguration of Spanning Trees with Many or Few Leaves

TL;DR

This paper studies the complexity of transforming one spanning tree into another through edge flips while maintaining constraints on the number of leaves, revealing PSPACE-completeness in many cases and polynomial cases for specific graph classes.

Contribution

It proves PSPACE-completeness for leaf constraints in spanning tree transformations and identifies graph classes where the problem is polynomial-time solvable.

Findings

Transformations with at most k leaves are PSPACE-complete for fixed k ≥ 3.

Transformations with at least k leaves are PSPACE-complete even on restricted graph classes.

Polynomial-time solutions exist for cographs, interval graphs, and when k=n-2.

Abstract

Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem of determining, given two spanning trees $T_1,T_2$ with an additional property $\Pi$, if there exists an edge flip transformation from $T_1$ to $T_2$ keeping property $\Pi$ all along. First we show that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at most $k$ (for any fixed $k \ge 3$) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at least $k$ leaves (where $k$ is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when $k=n-2$.