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 of maximum degree three and arbitrarily large girth. 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).

翻译：在没有共同端点的图形中, 匹配是一组匹配的边缘。 如果以美元表示的边缘端端点的诱导子图是环状的, 匹配的美美元是环状的。 如果以美元表示的边缘端端点的诱导子图是环状的, 则匹配的美元是环状的。 如果以美元表示的边缘端点是环状的, 则匹配的美元是环状的。 如果以美元表示的边缘端点的诱导子图是环状的, 则匹配的美元是环状的。 如果以美元表示的, 则匹配的问题仍然是环状的。 以美元表示的平面的直线图和每个正面的平面图( $C_ 4美元表示的平面图和每个正面图) 问题仍然是NP的。 此外, 我们研究问题的参数复杂性。 特别是, 我们证明问题在于双面图上与以美元表示的方块块块块图有W[1]- 硬度。 另一方面, 问题是固定的参数, 与以美元表示, $k$4$- 非平面图和每个平面图包括正面图的平面图和每个平面图。