A matching is a set of edges in a graph with no common endpoint. A matching $M$ is called acyclic if the induced subgraph on the endpoints of the edges in $M$ is acyclic. Given a graph $G$ and an integer $k$, Acyclic Matching Problem seeks for an acyclic matching of size $k$ in $G$. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs with maximum degree three and girth of arbitrary large. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter $k$. On the other hand, the problem is fixed parameter tractable with respect to $k$, for line graphs, $C_4$-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

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关于非循环匹配问题的参数化复杂度

匹配是图中没有公共端点的一组边。如果 $M$ 中边端点上的诱导子图是无环的，则匹配的 $M$ 称为无环。给定一个图 $G$ 和一个整数 $k$，非循环匹配问题在 $G$ 中寻找大小为 $k$ 的非循环匹配。已知该问题是 NP 完全的。在本文中，我们从不同方面研究了问题的复杂性。首先，我们证明对于最大度数为 3 且周长为任意大的平面二分图类，该问题仍然是 NP 完全的。此外，对于最大度数为 4 的平面线图类，该问题仍然是 NP 完全的。此外，我们研究了问题的参数化复杂性。特别地，我们证明了关于参数 $k$ 在二部图上的问题是 W[1]-hard。另一方面，