Algorithmic graph minor theory: improved grid minor bounds and wagner's contraction

  • Authors:

  • Erik D. Demaine;MohammadTaghi Hajiaghayi;Ken-ichi Kawarabayashi

  • Affiliations:

  • MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA;MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA;Graduate School of Information Sciences, Tohoku University, Sendai, Miyagi, Japan

  • Venue:
  • ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
  • Year:
  • 2006

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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, K3,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.