Independent Set Reconfiguration Under Bounded-Hop Token

TL;DR

This work introduces the $k$-Jump reconfiguration model, unifying Token Sliding ($k=1$) and Token Jumping (full diameter) and analyzes how rule changes affect ISReconf complexity. It proves that for connected graphs and $k \ge 3$, ISReconf under the $k$-Jump model is equivalent to the TJ model, thereby aligning reconfigurability power with TJ in this regime. It then provides a polynomial-time algorithm for $2$-ISReconf on split graphs, showing a separation from the $1$-Jump case where ISReconf is PSPACE-complete. Finally, it establishes NP-completeness of the Shortest-ISReconf under the $k$-Jump model for chordal graphs with diameter at most $2k+1$, indicating nuanced complexity for optimization variants even when decision variants align. Overall, the paper reveals a rich complexity landscape across $k$, graph classes, and whether the objective is existence or shortest sequences, highlighting new directions for reconfiguration problems.

Abstract

The independent set reconfiguration problem (ISReconf) is the problem of determining, for given independent sets I_s and I_t of a graph G, whether I_s can be transformed into I_t by repeatedly applying a prescribed reconfiguration rule that transforms an independent set to another. As reconfiguration rules for the ISReconf, the Token Sliding (TS) model and the Token Jumping (TJ) model are commonly considered. While the TJ model admits the addition of any vertex (as far as the addition yields an independent set), the TS model admits the addition of only a neighbor of the removed vertex. It is known that the complexity status of the ISReconf differs between the TS and TJ models for some graph classes. In this paper, we analyze how changes in reconfiguration rules affect the computational complexity of reconfiguration problems. To this end, we generalize the TS and TJ models to a unified reconfiguration rule, called the k-Jump model, which admits the addition of a vertex within distance k from the removed vertex. Then, the TS and TJ models are the 1-Jump and D(G)-Jump models, respectively, where D(G) denotes the diameter of a connected graph G. We give the following three results: First, we show that the computational complexity of the ISReconf under the k-Jump model for general graphs is equivalent for all k >= 3. Second, we present a polynomial-time algorithm to solve the ISReconf under the 2-Jump model for split graphs. We note that the ISReconf under the 1-Jump (i.e., TS) model is PSPACE-complete for split graphs, and hence the complexity status of the ISReconf differs between k = 1 and k = 2. Third, we consider the optimization variant of the ISReconf, which computes the minimum number of steps of any transformation between Is and It. We prove that this optimization variant under the k-Jump model is NP-complete for chordal graphs of diameter at most 2k + 1, for any k >=3.

Figures (10)

• Figure 1: A transformation $\langle I_0, I_1, \ldots, I_5 \rangle$ of independent sets between $I_s = I_0$ and $I_t = I_5$ under the TJ model, where tokens (i.e., the vertices in an independent set) are colored with gray. Note that there is no transformation between $I_s$ and $I_t$ under the TS model.
• Figure 2: No instance for split graphs under the $2$-Jump model. Note that this is a yes-instance under the $k$-Jump model, $k \ge 3$.
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Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Theorem 3
• Lemma 1
• proof
• Theorem 3
• Lemma 2