Summary: An induced matching in a graph \(G=(V,E)\) is a matching \(M\) such that \((V,M)\) is an induced subgraph of \(G\). Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al. [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least \(\frac{n}{40}\) and that there are twinless planar graphs that do not contain an induced matching of size greater than \(\frac{n}{27} + O(1)\). We improve both these bounds to \(\frac{n}{28} +O(1)\), which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size \(k\) has a kernel of size at most \(28k\). We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity.
Kanj et al. also presented an algorithm which decides in \(O(2^{159\sqrt{k}} + n)\)-time whether an \(n\)-vertex planar graph contains an induced matching of size \(k\). Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient \(O(2^{25.5\sqrt{k}} + n)\)-time algorithm. Its main ingredient is a new, \(O^{*}(4^l)\)-time algorithm for finding a maximum induced matching in a graph of branch width at most \(l\).