On the Complexity of Distance-$d$ Independent Set Reconfiguration

TL;DR

We study Distance-$d$ Independent Set Reconfiguration (D$d$ISR) for fixed $d\ge3$ under Token Sliding (TS) and Token Jumping (TJ). By linking D$d$IS to IS on graph powers $G^{d-1}$ and employing a suite of reductions (from ISR, 3-SAT Reconfiguration, and NCL), we classify the problem’s complexity across key graph classes. Notably, on chordal graphs TJ yields a polynomial-time algorithm for even $d$ and PSPACE-completeness for odd $d$, while on split graphs a TS-based dichotomy exists between $d=2$ and $d=3$ (with TJ showing a complementary dichotomy). We extend many $d=2$ hardness results to $d\ge3$ for perfect and planar graphs, provide a general TJ reduction on general graphs, and establish polynomial-time solvability on graphs with bounded diameter components (e.g., cographs) and related classes. The paper also outlines open problems for trees, highlighting a path toward a complete dichotomy and the role of rigid tokens and caterpillars as future directions.

Abstract

For a fixed positive integer $d \geq 2$, a distance-$d$ independent set (D$d$IS) of a graph is a vertex subset whose distance between any two members is at least $d$. Imagine that there is a token placed on each member of a D$d$IS. Two D$d$ISs are adjacent under Token Sliding ($\mathsf{TS}$) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping ($\mathsf{TJ}$), the target vertex needs not to be adjacent to the original one. The Distance-$d$ Independent Set Reconfiguration (D$d$ISR) problem under $\mathsf{TS}/\mathsf{TJ}$ asks if there is a corresponding sequence of adjacent D$d$ISs that transforms one given D$d$IS into another. The problem for $d = 2$, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of D$d$ISR on different graphs under $\mathsf{TS}$ and $\mathsf{TJ}$ for any fixed $d \geq 3$. On chordal graphs, we show that D$d$ISR under $\mathsf{TJ}$ is in $\mathtt{P}$ when $d$ is even and $\mathtt{PSPACE}$-complete when $d$ is odd. On split graphs, there is an interesting complexity dichotomy: D$d$ISR is $\mathtt{PSPACE}$-complete for $d = 2$ but in $\mathtt{P}$ for $d=3$ under $\mathsf{TS}$, while under $\mathsf{TJ}$ it is in $\mathtt{P}$ for $d = 2$ but $\mathtt{PSPACE}$-complete for $d = 3$. Additionally, certain well-known hardness results for $d = 2$ on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for $d \geq 3$.

Figures (8)

• Figure 1: Construction of a D$d$ISR's instance $(G, I, J, \sfTS)$ satisfying Proposition \ref{['prop:TS-power-graph']}. Paths $P_1, \dots, P_k$ are marked by light gray boxes. Paths $Q_{v_i^\star, v_j}$ and $Q_{w_i^\star, w_j}$ of length $d-1$ are represented by bold dashed egdes, where $1 \leq i, j \leq k$ and $i \neq j$. Tokens in $I$ (resp. $J$) are marked with black (resp. gray) color.
• Figure 2: An example of constructing a chordal graph $\Gp$ from $G$ for $d = 5$. Vertices in a light-gray box form a clique. Vertices in $V(\Gp) - V(G)$ are marked with the gray color.
• Figure 3: An example of constructing $\Gp$ from a given graph $G$ for some values of $d \in \{2,3,4\}$ (in case $d = 2$, we have $\Gp = G$). Vertices in $V(\Gp) - V(G)$ are marked with the gray color.
• Figure 4: An example of constructing the graphs $G$ and $\Gp$ from a 3SAT formula $\varphi$ used in ItoDHPSUU11 having three variables $x_1$, $x_2$, and $x_3$ and three clauses $c_1 = \{x_1, \overline{x_2}\}$, $c_2 = \{\overline{x_1}, x_2, x_3\}$, and $c_3 = \{\overline{x_2}, \overline{x_3}\}$. Paths of length $d-1$ are represented by dotted edges.
• Figure 5: An example of constructing the perfect graphs $\Gp$ and $\Gpp$ from $\Gtil$. Vertices in a light-gray box form a clique. Vertices in $V(\Gpp) - V(\Gp)$ are marked with the gray color.

Theorems & Definitions (25)

• proposition 1
• corollary 1
• proposition 2
• proof
• proposition 3
• proof
• corollary 2
• proof
• corollary 3
• theorem 1