Improved induced matchings in sparse graphs

  • Authors:

  • Rok Erman;Łukasz Kowalik;Matja Krnc;Tomasz Waleń

  • Affiliations:

  • Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia;Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland;Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia;Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

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 n40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n27+O(1). We improve both these bounds to n28+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(2^1^5^9^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(2^2^5^.^5^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.