Revisiting Token Sliding on Chordal Graphs

TL;DR

This paper investigates the computational complexity of the token sliding reconfiguration problem on chordal graphs, proving it is NP-hard even under certain restrictions and exploring its parameterized complexity.

Contribution

It establishes NP-hardness for token sliding connectivity on chordal graphs with bounded maximum clique-tree degree and analyzes the problem's complexity with respect to leafage and related parameters.

Findings

NP-hardness for maximum clique-tree degree d=4

W[1]-hardness with respect to leafage

Similar complexity results for token sliding reachability

Abstract

In this article, we revisit the complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs. In the \textsc{Token Sliding-Connectivity} problem, the input is a graph $G$ and an integer $k$, and the objective is to determine whether the reconfiguration graph $TS_k(G)$ of $G$ is connected. The vertices of $TS_k(G)$ are $k$-independent sets of $G$, and two vertices are adjacent if and only if one can transform one of the two corresponding independent sets into the other by sliding a vertex (also called a \emph{token}) along an edge. Bonamy and Bousquet [WG'17] proved that the \textsc{Token Sliding-Connectivity} problem is polynomial-time solvable on interval graphs but \NP-hard on split graphs. In light of these two results, the authors asked: can we decide the connectivity of $TS_k(G)$ in polynomial time for chordal graphs with \emph{maximum clique-tree degree} $d$? We answer this question in the negative and prove that the problem is \para-\NP-hard when parameterized by $d$. More precisely, the problem is \NP-hard even when $d = 4$. We then study the parameterized complexity of the problem for a larger parameter called \emph{leafage} and prove that the problem is \co-\W[1]-hard. We prove similar results for a closely related problem called \textsc{Token Sliding-Reachability}. In this problem, the input is a graph $G$ with two of its $k$-independent sets $I$ and $J$, and the objective is to determine whether there is a sequence of valid token sliding moves that transform $I$ into $J$.