Reconfiguring Multiple Connected Components with Size Multiset Constraints

TL;DR

CCR generalizes ISR by allowing connected components with sizes forming a multiset $\\mathcal{M}$ and introduces component jumping (CJ) and component sliding (CS) rules that treat components as transferable tokens with fixed sizes. While CCR is PSPACE-complete in general, the paper presents linear-time algorithms for CCR-CS on path graphs and for CCR-CS on cographs, as well as quadratic-time solvability for CCR-CJ on paths and linear-time solvability for CCR-CJ on chordal graphs when all components are equal-sized. The results reveal a meaningful difference between CJ and CS and extend ISR insights to a broader reconfiguration framework, with implications for systems where grouped resources (connected components) must be moved without altering their sizes. The work combines combinatorial reconfiguration with graph-structure techniques (buffers and inversions on paths, cotrees on cographs, CC-Piran graphs on chordal graphs) to establish tractability boundaries for CCR.

Abstract

We propose a novel generalization of Independent Set Reconfiguration (ISR): Connected Components Reconfiguration (CCR). In CCR, we are given a graph $G$, two vertex subsets $A$ and $B$, and a multiset $\mathcal{M}$ of positive integers. The question is whether $A$ and $B$ are reconfigurable under a certain rule, while ensuring that each vertex subset induces connected components whose sizes match the multiset $\mathcal{M}$. ISR is a special case of CCR where $\mathcal{M}$ only contains 1. We also propose new reconfiguration rules: component jumping (CJ) and component sliding (CS), which regard connected components as tokens. Since CCR generalizes ISR, the problem is PSPACE-complete. In contrast, we show three positive results: First, CCR-CS and CCR-CJ are solvable in linear and quadratic time, respectively, when $G$ is a path. Second, we show that CCR-CS is solvable in linear time for cographs. Third, when $\mathcal{M}$ contains only the same elements (i.e., all connected components have the same size), we show that CCR-CJ is solvable in linear time if $G$ is chordal. The second and third results generalize known results for ISR and exhibit an interesting difference between the reconfiguration rules.

Figures (3)

• Figure 1: A reconfiguration sequence in CCR-CJ. The start and target configurations $A$ and $B$ are shown in the upper left and lower right, respectively. $\mathcal{M}$ consists of one 2 and one 3. Black and gray vertices are in vertex subsets. Note that the upper left configuration and the lower left configuration are not adjacent under CJ because they exchange vertices between different connected components, which is allowed in TJ and TS. The reconfiguration sequence is also valid under CS.
• Figure 2: Relation between the reconfiguration rules. A solid arrow $\textsf{R} \longrightarrow \textsf{R}'$ means that, if the answer of CCR-R is YES, the answer of CCR-$\textsf{R}'$ is also YES.
• Figure 3: The CC-Piran graph such that the greedy algorithm fails. The solid upper and lower circles are components in $\mathcal{C}(A) \setminus \mathcal{C}(B)$ and $\mathcal{C}(B) \setminus \mathcal{C}(A)$, respectively. The dotted circle indicates a connected component with $k$ vertices in $G$ outside the CC-Piran graph. In this instance, we can reconfigure $A$ into $B$ by first moving the token of size $k$ to the right space, next moving the token of size 1, and then moving the token of size $k$ to the lower right space.

Theorems & Definitions (25)

• definition thmcounterdefinition: CCR
• definition thmcounterdefinition: CJ, CS, CS1
• theorem thmcountertheorem: *
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem