Directed Token Sliding

Reconfiguration problems involve determining whether two given configurations can be transformed into each other under specific rules. The Token Sliding problem asks whether, given two different set of tokens on vertices of a graph $G$, we can transform one into the other by sliding tokens step-by-step along edges of $G$ such that each resulting set of tokens forms an independent set in $G$. Recently, Ito et al. [MFCS 2022] introduced a directed variant of this problem. They showed that for general oriented graphs (i.e., graphs where no pair of vertices can have directed edges in both directions), the problem remains $\mathsf{PSPACE}$-complete, and is solvable in polynomial time on oriented trees. In this paper, we further investigate the Token Sliding problem on various oriented graph classes. We show that the problem remains $\mathsf{PSPACE}$-complete for oriented split graphs, bipartite graphs and bounded treewidth graphs. Additionally, we present polynomial-time algorithms for solving the problem on oriented cycles and cographs.

💡 Research Summary

The paper investigates the directed variant of the Token Sliding reconfiguration problem, where tokens placed on vertices of a graph must always form an independent set and may be moved only along directed edges. Building on the recent work of Ito et al., which showed PSPACE‑completeness for general oriented graphs and polynomial‑time solvability on oriented trees, the authors explore a range of more specific oriented graph classes.

First, they prove that the problem remains PSPACE‑complete on oriented split graphs. Starting from a known PSPACE‑complete instance on undirected split graphs, they construct an oriented split graph G′ by duplicating the clique part into two copies (K₁ and K₂), adding two special vertices c₁ and c₂, and directing edges so that every vertex in K₁∪K₂ has arcs to c₁ and from c₂. The original independent‑set vertices are copied once. A careful sequence of token slides through c₁→c₂ simulates any slide in the original graph, while the reverse direction uses a longer detour that forces tokens to move back through the duplicated clique. Two lemmas establish a bijective correspondence between reconfiguration sequences of the original and the constructed instance, yielding PSPACE‑hardness for oriented split graphs.

Second, the authors extend PSPACE‑completeness to oriented bipartite graphs. For each vertex v of an undirected bipartite graph G they create a copy v′, and for each undirected edge uv they choose an orientation (u→v or v→u). Then they complete a directed 4‑cycle (u→v, v→u′, u′→v′, v′→u). This gadget forces any token move between original vertices to be simulated by a short three‑step walk through the copies, preserving independence at each step. Lemma 3 shows that each vertex and its copy have identical neighborhoods, which is crucial for the correctness of the simulation. Lemmas 4 and 5 prove the forward and backward reductions, establishing PSPACE‑completeness for oriented bipartite graphs.

Third, the paper notes that the same reduction technique works for graphs of bounded treewidth: by orienting each edge arbitrarily, the bounded‑treewidth property is preserved, and the known PSPACE‑completeness for the undirected case carries over.

On the algorithmic side, the authors present polynomial‑time algorithms for two classes where the directed reconfiguration graph becomes tractable. For oriented cycles, they exploit the cyclic order of vertices. By representing a configuration as a binary string of token positions and using dynamic programming over the cycle, they can decide reachability in O(n³) time (or better with more refined techniques). For oriented cographs, they use the fact that cographs admit a cotree decomposition. The cotree is processed bottom‑up, maintaining for each node the set of feasible token placements consistent with the subgraph. Because each cotree node combines two induced subgraphs via disjoint union or join, the DP merges the feasible sets in polynomial time; the overall algorithm runs in time polynomial in the number of vertices and the treewidth of the cotree (which is bounded).

The paper concludes by summarizing the landscape: directed Token Sliding is PSPACE‑complete for a wide range of dense or bipartite oriented graphs, yet becomes efficiently solvable on highly structured sparse classes such as cycles and cographs. The authors suggest future work on other graph families (planar, bounded degree) and on alternative reconfiguration rules (token jumping, addition/removal) in the directed setting. This work significantly advances our understanding of asymmetric reconfiguration problems and delineates the boundary between intractable and tractable instances.