An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most $k$ vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size $k$ of the deletion set as the parameter. First, we prove a $6k$-vertex kernel for this problem, improving the previous result of $7k$. Second, we give an $O^*(1.6957^k)$-time and polynomial-space algorithm, improving the previous running-time bound of $O^*(1.7485^k)$.

翻译：若某个导出子图中每个顶点均为1度顶点，则称该子图为导出匹配。\textsc{几乎导出匹配}问题要求判定能否从输入图中删除至多$k$个顶点，使得剩余图构成一个导出匹配。本文以删除集大小$k$为参数研究该问题的参数化算法。首先，我们为该问题证明了一个$6k$顶点核，改进了此前$7k$的结果。其次，我们给出了一个时间复杂度$O^*(1.6957^k)$且空间复杂度为多项式的算法，将此前$O^*(1.7485^k)$的运行时间上界进行了改进。