Almost Induced Matching: Linear Kernels and Parameterized Algorithms.

The Almost Induced Matching problem asks whether we can delete at most k vertices from a graph such that the remaining graph is an induced matching, i.e., a graph with each vertex of degree﾿1. This paper studies parameterized algorithms for this problem by taking the size of deletion set k as the parameter. By using the techniques of finding maximal 3-path packings and an extended crown decomposition, we obtain the first linear vertex kernel for this problem, improving the previous quadratic kernel. We also present an $$O^*1.7485^k$$-time and polynomial-space algorithm, which is the best known parameterized algorithm for this problem.