On Reconfiguration Graphs of Independent Sets Under Token Sliding

Abstract

An independent set of a graph G is a vertex subset I such that there is no edge joining any two vertices in I. Imagine that a token is placed on each vertex of an independent set of G. The ${\textsf{TS}}$- (${\textsf{TS}}_k$-) reconfiguration graph of G takes all non-empty independent sets (of size k) as its nodes, where k is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph G: (1) Whether the ${\textsf{TS}}_k$-reconfiguration graph of G belongs to some graph class ${\mathcal {G}}$ (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If G satisfies some property ${\mathcal {P}}$ (including s-partitedness, planarity, Eulerianity, girth, and the clique’s size), whether the corresponding ${\textsf{TS}}$- (${\textsf{TS}}_k$-) reconfiguration graph of G also satisfies ${\mathcal {P}}$, and vice versa. Additionally, we give a decomposition result for splitting a ${\textsf{TS}}_k$-reconfiguration graph into smaller pieces.

Appendix A: Connectivity and Diameter of \({\textsf{TS}}_k(G)\) for Specific Graph Classes

For certain graph G, Table 3 includes some properties of the connectivity and diameter of ${\textsf{TS}}_k(G)$ that can be derived from known results.

Table 3 Connectivity and diameter of ${\textsf{TS}}_k(G)$ ($2 \le k \le \alpha (G)$). Here $n = \vert V(G) \vert$

• 1. In Kamiński et al. [12], showed the $\texttt{PSPACE}$-completeness of Independent Set Reconfiguration (ISR) under any of ${\textsf{TS}}, {\textsf{TJ}}$, or ${\textsf{TAR}}$ when the input graph is a perfect graph. Combining their reduction from Shortest Path Reconfiguration (SPR) and an example of a reconfiguration graph of SPR having exponentially large diameter in the size of the input graph [11] gives us an example of ${\textsf{TS}}_k(G)$ with exponentially large diameter in $n = \vert V(G) \vert$.

  Observe that one can construct a perfect graph G where ${\textsf{TS}}_k(G)$ is not connected. For instance, take G as the star $K_{1,n}$. Then, for $n \ge k+1$, ${\textsf{TS}}_k(G)$ is not connected. This also holds for other graph classes such as $P_4$-free graphs, trees, bipartite permutation graphs, and interval graphs.

• 2. They also showed that ${\textsf{TJ}}_k(G)$ is connected and its diameter is O(n) when G is a connected even-hole-free graph. Observe that when $k = \alpha (G)$, we have ${\textsf{TJ}}_k(G) \simeq {\textsf{TS}}_k(G)$.

• 3. Kamiński et al. [12] designed a linear-time algorithm that decides whether there is a path between $I, J \in {\textsf{TS}}_k(G)$, and if yes, outputs a shortest one, where G is $P_4$-free. One can verify that their algorithm indeed outputs a path in ${\textsf{TS}}_k(G)$ of length $O(n^2)$.

• 4. Bonsma et al. [5] show that when G is a connected claw-free graph, ${\textsf{TS}}_k(G)$ is always connected, and they provided a polynomial-time algorithm for outputting a path between any pair $I, J \in {\textsf{TS}}_k(G)$.

• 5–6. Demaine et al. [8] designed a linear-time algorithm for deciding, whether there is a path between $I, J \in {\textsf{TS}}_k(G)$, and if yes, output a path of length $O(n^2)$, provided that G is a tree. They also gave an example of an instance (G, I, J) where G is a path and the length of a shortest path between $I, J \in {\textsf{TS}}_k(G)$ is $\varOmega (n^2)$.

• 7. Fox-Epstein et al. [10] designed a cubic-time algorithm for deciding, whether there is a path between $I, J \in {\textsf{TS}}_k(G)$, and if yes, output a path of length $O(n^2)$, provided that G is a bipartite permutation graph.

• 8. Bonamy and Bousquet [4] designed a polynomial-time algorithm for deciding whether ${\textsf{TS}}_k(G)$ is connected when G is an interval graph. However, they did not provide any estimation on its diameter. Motivated by this question, Briański et al. [6] recently showed that the diameter of ${\textsf{TS}}_k(G)$ is $O(kn^2)$.