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DOI: 10.4230/LIPIcs.STACS.2010.2460
URN: urn:nbn:de:0030-drops-24605
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2460/
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Dorn, Frederic

Planar Subgraph Isomorphism Revisited

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Abstract

The problem of {\sc Subgraph Isomorphism} is defined as follows: Given a \emph{pattern} $H$ and a \emph{host graph} $G$ on $n$ vertices, does $G$ contain a subgraph that is isomorphic to $H$? Eppstein [SODA 95, J'GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order $k$, and arbitrary planar host graph, improving upon the $O(n^{\sqrt{k}})$-time algorithm when using the ``Color-coding'' technique of Alon et al [J'ACM 95]. Eppstein's algorithm runs in time $k^{O(k)} n$, that is, the dependency on $k$ is superexponential. We improve the running time to $2^{O(k)} n$, that is, single exponential in $k$ while keeping the term in $n$ linear. Next to deciding subgraph isomorphism, we can construct a solution and count all solutions in the same asymptotic running time. We may enumerate $\omega$ subgraphs with an additive term $O(\omega k)$ in the running time of our algorithm. We introduce the technique of ``embedded dynamic programming'' on a suitably structured graph decomposition, which exploits the number and topology of the underlying drawings of the subgraph pattern (rather than of the host graph).

BibTeX - Entry

@InProceedings{dorn:LIPIcs:2010:2460,
  author =	{Frederic Dorn},
  title =	{{Planar Subgraph Isomorphism Revisited}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{263--274},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Jean-Yves Marion and Thomas Schwentick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2460},
  URN =		{urn:nbn:de:0030-drops-24605},
  doi =		{10.4230/LIPIcs.STACS.2010.2460},
  annote =	{Keywords: Graph algorithms, Subgraph Isomorphism, NP-hard problems, Dynamic programming, Topological graph theory}
}

Keywords: Graph algorithms, Subgraph Isomorphism, NP-hard problems, Dynamic programming, Topological graph theory
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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