Excluding any graph as a minor allows a low tree-width 2-coloring

DeVos, Matt; Ding, Guoli; Oporowski, Bogdan; Sanders, Daniel P.; Reed, Bruce; Seymour, Paul; Vertigan, Dirk

doi:10.1016/j.jctb.2003.09.001

Cited by 90 publications

(77 citation statements)

References 8 publications

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“…This result is perhaps the ultimate in a series of contraction decompositions [26,27,13], and nicely parallels the deletion decomposition of H-minor-free graphs [15,12]. THEOREM 1.…”

Section: Our Resultssupporting

confidence: 59%

“…The idea is to partition the vertices or edges of the graph into a small number k of pieces such that deleting any one of the pieces results in a bounded-treewidth graph (where the bound depends on k). Such a decomposition is known for planar graphs [4], bounded-genus graphs [18], apexminor-free graphs [18], and H-minor-free graphs [15,12]. However, this decomposition approach is effectively limited to problems whose optimal solution only improves when deleting edges or vertices from the graph.…”

Section: Introductionmentioning

confidence: 99%

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Contraction decomposition in h-minor-free graphs and algorithmic applications

Demaine

Hajiaghayi

Kawarabayashi

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

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We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, generalizing previous such decompositions for planar and bounded-genus graphs, and solving the main open problem in this area (posed at SODA 2007). Our decomposition can be computed in polynomial time, resulting in a general framework for approximation algorithms, particularly PTASs (with k ≈ 1/ε), and fixed-parameter algorithms, for problems closed under contractions in graphs excluding a fixed minor. For example, our approximation framework gives the first PTAS for TSP in weighted H-minor-free graphs, solving a decade-old open problem of Grohe; and gives another fixed-parameter algorithm for k-cut in H-minor-free graphs, which was an open problem of Downey et al. even for planar graphs.To obtain our contraction decompositions, we develop new graph structure theory to realize virtual edges in the clique-sum decomposition by actual paths in the graph, enabling the use of the powerful Robertson-Seymour Graph Minor decomposition theorem in the context of edge contractions (without edge deletions). This requires careful construction of paths to avoid blowup in the number of required paths beyond 3. Along the way, we strengthen and simplify contraction decompositions for bounded-genus graphs, so that the partition is determined by a simple radial ball growth independent of handles, starting from a set of vertices instead of just one, as long as this set is tight in a certain sense. We show that this tightness property holds for a constant number of approximately shortest paths in the surface, introducing several new concepts such as dives and rainbows.

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“…This result is perhaps the ultimate in a series of contraction decompositions [26,27,13], and nicely parallels the deletion decomposition of H-minor-free graphs [15,12]. THEOREM 1.…”

Section: Our Resultssupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Contraction decomposition in h-minor-free graphs and algorithmic applications

Demaine

Hajiaghayi

Kawarabayashi

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

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“…The proof of this decomposition result is relatively simple, showing the power of our main decomposition result. An existential version of this result was shown by DeVos et al [22] using a complicated, and not obviously constructive, approach; here we show that a much simpler, and constructive, solution is possible using known results from an earlier paper of Grohe [30]. Even for the case k = 2, the result is very interesting: every H-minor-free graph is just the "sum" of two bounded-treewidth graphs.…”

Section: Introductionmentioning

confidence: 52%

Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring

Demaine¹,

Hajiaghayi²,

Kawarabayashi

46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)

154

200

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“…Tree-Width Tree-Depth 1 proper coloring 2 acyclic coloring [16] star coloring [54] p low tree-width decomposition [31] low tree-depth decomposition [84] Following [84], we will make use of the notation χ p (G) for the minimum number of colors need for a vertex coloring of G such that i < p parts induce a subgraph of tree-depth at most i. These graph invariants ("generalized chromatc numbers") form a non-decreasing sequence:…”

Section: Parametermentioning

confidence: 99%

Structural Properties of Sparse Graphs

Nešetřil¹

2008

Electronic Notes in Discrete Mathematics

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Excluding any graph as a minor allows a low tree-width 2-coloring

Cited by 90 publications

References 8 publications

Contraction decomposition in h-minor-free graphs and algorithmic applications

Contraction decomposition in h-minor-free graphs and algorithmic applications

Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring

Structural Properties of Sparse Graphs

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