Changing Induced Subgraph Isomorphisms Under Extended Reconfiguration Rules

TL;DR

ISIsoR extends reconfiguration by allowing up to $k$ simultaneous moves for tokens representing an $H$-induced subgraph, and the paper proves PSPACE-completeness for ISIsoR with constant $k$ under additive pattern classes, derives negative and positive meta-theorems parameterized by $\mu=|V(H)|-k$, and provides XP algorithms based on a clique-compressed reconfiguration graph. It also analyzes Independent Set Reconfiguration as a special case, showing hardness on perfect graphs and polynomial-time solvability on perfect graphs when parameterized by $\mu$, and establishes connections between $k$-TS ISR and standard TS on even-hole-free graphs. Collectively, the work clarifies how extending reconfiguration rules alters tractability boundaries and informs solver design for practical reconfiguration problems. The results map complexity transitions across graph classes and parameter regimes, guiding both theoretical understanding and algorithmic development in extended reconfiguration contexts.

Abstract

In a reconfiguration problem, we are given two feasible solutions of a combinatorial problem and our goal is to determine whether it is possible to reconfigure one into the other, with the steps dictated by specific reconfiguration rules. Traditionally, most studies on reconfiguration problems have focused on rules that allow changing a single element at a time. In contrast, this paper considers scenarios in which $k \ge 2$ elements can be changed simultaneously. We investigate the general reconfiguration problem of isomorphisms. For the Induced Subgraph Isomorphism Reconfiguration problem, we show that the problem remains $\textsf{PSPACE}$-complete even under stringent constraints on the pattern graph when $k$ is constant. We then give two meta-theorems applicable when $k$ is slightly less than the number of vertices in the pattern graph. In addition, we investigate the complexity of the Independent Set Reconfiguration problem, which is a special case of the Induced Subgraph Isomorphism Reconfiguration problem.

Figures (4)

• Figure 1: A sequence $\langle S_s=S_0, S_1, S_2=S_t \rangle$ of independent sets within the same graph cannot be achieved under either $\mathsf{TS}$ or $\mathsf{TJ}$, but it can be achieved under both $$2$\text{-}\mathsf{TS}$ and $$2$\text{-}\mathsf{TJ}$. Tokens corresponding to the independent sets are marked in black.
• Figure 2: An example of constructing $G$ from $W=(\Sigma,A)$ and $(w_s,w_t)$ such that $\Sigma=\{a,b,c,d\}$, $A=\{(a,b),(a,c),(b,b),(b,d),(c,a),(c,b),(c,d),(d,a),(d,c),(d,d)\}$, and $|w_s|=|w_t|=5$. Each node in the layers corresponds to a subgraph isomorphic to $tF$, and each double line represents all possible edges between the two subgraphs.
• Figure 3: Illustration of our construction of $G$. The double lines represent all possible edges between the two subgraphs. The black tokens are placed on the initial $H$-induced subgraph isomorphic set $S_s$.
• Figure 4: Illustration of our construction of $G'$. The double lines represent all possible edges between the two vertex sets. The dotted line between $A$ and $B_i$ represents all edges in $G^*$.

Theorems & Definitions (26)

• Theorem 1
• proof : Proof (Sketch)
• Corollary 2: $\ast$
• Theorem 3
• proof : Proof (Sketch)
• Corollary 4: $\ast$
• Theorem 5
• Proposition 6: $\ast$
• Corollary 7
• Corollary 8