Independent set reconfiguration in H-free graphs

TL;DR

This work studies Independent Set Reconfiguration (ISR) in $H$-free graphs under token sliding ($TS$) and token jumping ($TJ$). It extends Alekseev-type reductions to ISR by using a $t$-subdivision construction $G_t$ and canonical extensions to connect reconfiguration in $G$ with that in $G_t$, establishing a PSPACE-hardness dichotomy: $TS/TJ$ are $ extsf{PSPACE}$-complete unless $H$ is a path, a claw, or a subdivision of the claw. It then shows a positive result: Token Sliding can be solved in polynomial time on fork-free graphs, via a sequence of reductions to prime graphs, modular-decomposition-based analysis, and a cycle-resolution strategy that either yields a reconfiguration or identifies permanently blocked vertices. The results advance understanding of ISR in structurally constrained graphs and integrate modular decomposition, augmenting-path techniques, and claw-based structural characterizations to obtain both hardness and tractable cases. The work also outlines open questions about Token Jumping on fork-free graphs and the complexity for $P_5$-free MIS, highlighting avenues for further exploration.

Abstract

Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev's theorem, showing that the problem is PSPACE-complete unless $H$ is a path or a subdivision of the claw. We then show that under the token sliding rule, the problem admits a polynomial-time algorithm if the input graph is fork-free.

Figures (4)

• Figure 1: Small graphs of interest.
• Figure 2: An illustration in the two cases of the proof of Lemma \ref{['lem:first-neighbor-path']}. The dashed lines on the left indicate non-edges and the red squares indicate tokens.
• Figure 3: The fork-free prime expansions of a claw. The claws considered in Lemma \ref{['lem:rotation-claw']} are marked by thick edges. The graph $H_5$ is shown three times, once for each of its induced claws.
• Figure 4: An illustration of the situation in the proof of Theorem \ref{['thm:ts_tj_equiv_fork']}. The red squares represent the initial positions of the tokens. The dashed lines represent non-edges.

Theorems & Definitions (56)

• Theorem 1: alekseev1982effect
• Theorem 2
• Theorem 3
• Theorem 4: alekseev1982effect
• Theorem 5
• Claim 1
• proof
• Claim 2
• proof
• Claim 3