Revisiting Token Sliding on Chordal Graphs

TL;DR

The complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs is revisited and the parameterized complexity of the problem for a larger parameter calledleafage is studied and it is proved that the problem is \para-\NP-hard when parameterized by d.

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 $J$.

Figures (12)

• Figure 1: The reduced instance for the Token Sliding Connectivity problem parameterized by the maximum clique-tree degree and its corresponding clique-tree.
• Figure 2: The input Instance of TS-Reachability, $G$ is a split and $C$ is a clique graph.
• Figure 3: Reduction for Token Sliding Reachability problem parameterized by the maximum clique-tree degree and the corresponding clique-tree.
• Figure 4: (Top) Structure of the model tree. Node $t_0$ is the central vertex and $t_1, t_2, \dots, t_k$ and $t_p$ are its children. The structure between $t_0$ and $t_p$ is to park the tokens. (Bottom) The orange vertex denotes the conditional free pass between $T_i$ and $T_j$.
• Figure 5: Add $(n-1)$ pink vertices for each $t_i$ where $i\in[k]$ as follows: Each pink vertex $y_i^p$ intersects $\{u_i^p,u_i^{p+1},\ldots,u_i^n\}$ and $\{w_i^1,w_i^2,\ldots,w_i^{p+1}\}$. In the figure $n$ is taken to be $5$. No token from $T_i$ can be moved to a pink vertex.

Theorems & Definitions (16)

• Proposition : Theorem $2$ and $3$ in DBLP:conf/wg/BonamyB17
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Claim 8