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).
翻译:在没有共同端点的图形中, 匹配是一组边缘。 如果以美元表示的边缘端端端点的诱导子图是单向的, 匹配的美美元是一个环曲。 如果以美元表示的边缘端端点上的诱导子图是单向的, 则匹配的美元是单向的。 如果以美元表示的边缘端点上的诱导子图是单向的, 则匹配的美元是单向的。 如果以美元表示的, 则匹配的美元是单向的。 如果以美元表示, 则匹配的美元是单向的。 如果以美元表示, 则匹配的美元是单向的双向图形。 此外, 单向匹配的问题是以美元表示的复杂度, 特别是, 我们证明双向方图上的问题是W[ 1] - 硬的, 问题在于参数 。 另一方面, 问题在于$k$$$( 、 $_ $_ $_ 4- freflefleast ligal gramaps and every prial- prial grame) grame- pain 和每个平面图的平面图和每个平面图的平面图( las- gram- degram- degram- degram- drealbrealblock- grame) grame- grame- grame- grame- grame- graphalgalgalgalgalgalgalgmapsgmapsgmapsgmaps) 。