Planar Subgraph Isomorphism Revisited

Author Frederic Dorn



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Frederic Dorn

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Frederic Dorn. Planar Subgraph Isomorphism Revisited. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 263-274, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)
https://doi.org/10.4230/LIPIcs.STACS.2010.2460

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).
Keywords
  • Graph algorithms
  • Subgraph Isomorphism
  • NP-hard problems
  • Dynamic programming
  • Topological graph theory

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