AI 中文总结
研究毛毛虫图上组件跳跃时连通分量重配置问题（\(\CCRCJ\)）的复杂度与算法，证明其在毛毛虫图上NP难，解决弦图多重集大小约束下复杂度问题，还将路径图上判定算法从\(O(n^2)\)改进为\(O(n\log n)\)，并给出大空空间下重配置序列。
AI 中文摘要
我们研究连通分量重配置问题（CCR），其中图上的连通分量根据指定的重配置规则进行变换。CCR通过将令牌视为规定大小的连通分量而非单个顶点来推广独立集重配置。在CCR的变体中，我们聚焦于组件跳跃模型\(\CCRCJ\)。Nakahata引入此问题并表明，对于任意组件大小，\(\CCRCJ\)的判定问题在路径图上可在\(O(n^2)\)时间内解决，且当所有连通分量大小相同时在弦图上可在多项式时间内解决。但弦图在多重集大小约束下的复杂度仍未解决。本文从复杂度理论和算法角度研究此多重集版本的\(\CCRCJ\)。首先，我们证明\(\CCRCJ\)即使在毛毛虫图（树和弦图中非常受限的子类，仅比路径图稍复杂）上也是NP难的，这解决了Nakahata关于弦图在多重集大小约束下的开放问题。其次，我们重新审视路径图上的\(\CCRCJ\)，将判定问题的先前\(O(n^2)\)时间算法改进为\(O(n\log n)\)时间算法。此外，当实例有足够大的空空间时，我们表明存在长度为\(O(n\log n)\)的重配置序列且可有效输出。
英文摘要
We study the Connected Components Reconfiguration problem (CCR), in which connected components on a graph are transformed according to a specified reconfiguration rule. CCR generalizes Independent Set Reconfiguration by treating tokens not as individual vertices but as connected components of prescribed sizes. Among the variants of CCR, we focus on the component-jumping model, denoted by \CCRCJ. Nakahata.\ introduced this problem and showed that the decision problem for \CCRCJ~can be solved in $O(n^2)$ time on path graphs for arbitrary component sizes, and in polynomial time on chordal graphs when all connected components have the same size. However, the complexity on chordal graphs under a multiset size constraint remained open. In this paper, we study this multiset version of \CCRCJ~from both complexity-theoretic and algorithmic viewpoints. First, we prove that \CCRCJ~is NP-hard even on caterpillar graphs, which is a very restricted subclass of trees and chordal graphs minimally above path graphs. This result immediately implies NP-hardness for chordal graphs under a multiset size constraint, thereby resolving Nakahata's open problem on chordal graphs under multiset size constraints. Second, we revisit \CCRCJ~on path graphs. We improve the previous $O(n^2)$-time algorithm for the decision problem by giving an $O(n\log n)$-time decision algorithm. Moreover, when the instance has sufficiently large empty space, we show that there exists a reconfiguration sequence of length $O(n\log n)$, and such a sequence can be output efficiently.
Comments28 pages,6 figures
