Shortest Reconfiguration Sequence for Sliding Tokens on Spiders

TL;DR

This paper proves that finding the shortest reconfiguration sequence for sliding tokens on spider graphs can be done in polynomial time, unlike the general NP-hard case.

Contribution

It establishes a polynomial-time algorithm for shortest reconfiguration sequences specifically on spider graphs, a class of trees with a single high-degree vertex.

Findings

Polynomial-time algorithm for spiders

Shortest sequence computation is feasible on specific graph classes

Advances understanding of reconfiguration problems

Abstract

Suppose that two independent sets $I$ and $J$ of a graph with $\vert I \vert = \vert J \vert$ are given, and a token is placed on each vertex in $I$. The Sliding Token problem is to determine whether there exists a sequence of independent sets which transforms $I$ into $J$ so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. It is one of the representative reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. For a yes-instance of a reconfiguration problem, finding a shortest reconfiguration sequence has a different aspect. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be $\mathsf{NP}$-hard to find a shortest sequence between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for spiders (i.e., trees having exactly one vertex of degree at least three).