**Reference**:- Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi, “Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring”, in
*Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005)*, Pittsburgh, PA, October 23–25, 2005, pages 637–646. **Abstract**:- At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomial-time algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into bounded-genus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmic-approximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor.
**Length**:- The paper is 10 pages.
**Availability**:- The paper is available in PostScript (314k), gzipped PostScript (110k), and PDF (177k).
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Last updated October 17, 2021 by Erik Demaine.