Maximum induced matchings close to maximum matchings

Extending results of Kobler and Rotics (2003), Cameron and Walker (2005) gave a complete structural description of the graphs G   where the matching number ν(G)ν(G) equals the induced matching number ν2(G)ν2(G). We present a short proof of their result and use it to study graphs G   with ν(G)−ν2(G)≤kν(G)−ν2(G)≤k. We show that the recognition of these graphs can be done in polynomial time for fixed k, and is fixed parameter tractable when parameterized by k for graphs of bounded maximum degree.