AI 中文总结
本文证明，判断最大度为3的二分图中相关完美匹配强制集、反强制集存在性的多个问题均为NP完全，填补了反强制集相关问题在该图类下的复杂性空白。
AI 中文摘要
在图$G$中，若存在唯一完美匹配$M$使得$F \subseteq M$，则边集$F$被称为强制集；类似地，若边集为$E(G)\setminus A$的图具有唯一完美匹配，则边集$A$被称为反强制集。已知给定最大度为3的二分图$G$及完美匹配$M$，判断是否存在大小不超过$k$的$M$的强制集的问题是NP完全的；此外，给定最大度为4的二分图$G$及完美匹配$M$，判断是否存在大小不超过$k$的$M$的反强制集的问题也是NP完全的。进一步地，给定最大度为5的二分图，判断是否存在可通过大小不超过$k$的强制集使其唯一的完美匹配$M$的问题同样是NP完全的。相比之下，判断是否存在可通过大小不超过$k$的反强制集使其唯一的完美匹配$M$的计算复杂性，即使对于一般图也尚未明确。本文证明，所有上述问题在限制为最大度为3的二分图时，仍然保持NP完全性。
英文摘要
In a graph $G$, a set of edges $F$ is called a \emph{forcing set} if there exists a unique perfect matching $M$ such that $F \subseteq M$. Similarly, a set of edges $A$ is called an \emph{anti-forcing set} if the graph with edge set $ E(G)\setminus A$ has a unique perfect matching. It is known that, given a bipartite graph $G$ of maximum degree~$3$ and a perfect matching $M$, the problem of deciding whether there exists a forcing set of size at most $k$ for $M$ is NP-complete. Moreover, given a bipartite graph $G$ of maximum degree~$4$ and a perfect matching $M$, the problem of deciding whether there exists an anti-forcing set of size at most $k$ for $M$ is NP-complete. Furthermore, given a bipartite graph of maximum degree~$5$, the problem of deciding whether there exists a perfect matching $M$ that can be made unique by a forcing set of size at most $k$ is also NP-complete. In contrast, the computational complexity of deciding whether there exists a perfect matching $M$ that can be made unique by an anti-forcing set of size at most $k$ is not known, even for general graphs. In this paper, we show that all of these problems remain NP-complete even when restricted to bipartite graphs of maximum degree~$3$.
Comments9 pages, 5 figures
