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Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

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Algorithms and Computation (ISAAC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4288))

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Abstract

We explore the three main avenues of research still unsolved in the algorithmic graph-minor theory literature, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties.

First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K 3,k-minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixed-parameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters. Third, we disprove a variation of Wagner’s Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.

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Author information

Authors and Affiliations

  1. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA, 02139, USA

    Erik D. Demaine & MohammadTaghi Hajiaghayi

  2. Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    MohammadTaghi Hajiaghayi

  3. Graduate School of Information Sciences, Tohoku University, Aramaki aza Aoba 09, Aoba-ku, Sendai, Miyagi, 980-8579, Japan

    Ken-ichi Kawarabayashi

Authors
  1. Erik D. Demaine
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  2. MohammadTaghi Hajiaghayi
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  3. Ken-ichi Kawarabayashi
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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Japan

    Tetsuo Asano

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© 2006 Springer-Verlag Berlin Heidelberg

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Demaine, E.D., Hajiaghayi, M., Kawarabayashi, Ki. (2006). Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_3

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  • DOI: https://doi.org/10.1007/11940128_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49694-6

  • Online ISBN: 978-3-540-49696-0

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