Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint

TL;DR

This paper studies the complexity of transforming one spanning tree into another with constraints on degree or diameter, revealing polynomial solvability and hardness results for various cases.

Contribution

It provides a detailed complexity analysis of spanning tree reconfiguration under degree and diameter constraints, identifying which problems are polynomial-time solvable or NP-hard.

Findings

Lower bound on maximum degree is polynomial-time solvable.

Upper bound on maximum degree is PSPACE-complete.

Upper bound on diameter is polynomial-time solvable.

Abstract

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.