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

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Abstract

We explore three important avenues of research in algorithmic graph-minor theory, 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, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. 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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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. National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Ken-ichi Kawarabayashi

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  1. Erik D. Demaine
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  2. MohammadTaghi Hajiaghayi
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  3. Ken-ichi Kawarabayashi
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Demaine, E.D., Hajiaghayi, M. & Kawarabayashi, Ki. Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction. Algorithmica 54, 142–180 (2009). https://doi.org/10.1007/s00453-007-9138-y

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