# Local Tree-Width, Excluded Minors, and Approximation Algorithms

@article{Grohe2003LocalTE,
title={Local Tree-Width, Excluded Minors,
and Approximation Algorithms},
author={Martin Grohe},
journal={Combinatorica},
year={2003},
volume={23},
pages={613-632}
}
• Martin Grohe
• Published 2003
• Mathematics, Computer Science
• Combinatorica
The local tree-width of a graph G=(V,E) is the function ltwG :ℕ→ℕ that associates with every r∈ℕ the maximal tree-width of an r-neighborhood in G. Our main grapht heoretic result is a decomposition theorem for graphs with excluded minors, which says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width.As an application of this theorem, we show that a number of combinatorial optimization problems, suchas Minimum Vertex Cover, Minimum Dominating Set, and… Expand
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#### References

SHOWING 1-10 OF 34 REFERENCES
Diameter and Treewidth in Minor-Closed Graph Families
• D. Eppstein
• Mathematics, Computer Science
• Algorithmica
• 2000
It is shown that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family, and the O(D) bound above can be extended to bounded-genus graphs. Expand
NC-Algorithms for Graphs with Small Treewidth
A parallel algorithm for recognizing graphs with treewidth ≤ k, for constant k, and building the corresponding tree-decomposition, that uses O(log n) time and O(n3k+4) processors on a CRCW PRAM is given. Expand
A separator theorem for graphs with an excluded minor and its applications
• Mathematics, Computer Science
• STOC '90
• 1990
It follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and solve any sparse system of n linear equations in n unknowns in time O(n). Expand
Deciding First-Order Properties of Locally Tree-Decomposalbe Graphs
• Mathematics, Computer Science
• ICALP
• 1999
It is shown that for each locally tree-decomposable class C of graphs and for each property φ of graphs that is definable in first-order logic, there is a linear time algorithm deciding whether a given graph G ∈ C has property ϕ. Expand
Easy Problems for Tree-Decomposable Graphs
• Computer Science, Mathematics
• J. Algorithms
• 1991
Abstract Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets canExpand
Graph Minors. XVI. Excluding a non-planar graph
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2003
It is found that every graph with no minor isomorphic to L may be constructed by piecing together in a tree-structure graphs each of which "almost" embeds in some surface in which L cannot be embedded. Expand
Graph minors. III. Planar tree-width
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1984
It is proved that for any fixed planar graph H, every planar graphs with sufficiently large tree-width has a minor isomorphic to H. Expand
Approximation algorithms for NP-complete problems on planar graphs
• B. Baker
• Mathematics, Computer Science
• 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
• 1983
A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand
A Separator Theorem for Planar Graphs
• Mathematics
• 1977
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains moreExpand
A polynomial-time approximation scheme for weighted planar graph TSP
• Mathematics, Computer Science
• SODA '98
• 1998
This work finds a salesman tour of total cost at most (1 + E) times optimal in time n for any E > 6, and presents a quasi-polynomial time algorithm for the Steiner version of this problem. Expand