# Diameter and Treewidth in Minor-Closed Graph Families, Revisited

@article{Demaine2004DiameterAT, title={Diameter and Treewidth in Minor-Closed Graph Families, Revisited}, author={E. Demaine and M. Hajiaghayi}, journal={Algorithmica}, year={2004}, volume={40}, pages={211-215} }

Abstract
Eppstein [5] characterized the minor-closed graph families for which the treewidth is bounded
by a function of the diameter, which includes, e.g., planar graphs. This characterization has been used as the
basis for several (approximation) algorithms on such graphs (e.g., [2] and [5]–[8]). The proof of Eppstein
is complicated. In this short paper we obtain the same characterization with a simple proof. In addition, the
relation between treewidth and diameter is slightly better and… Expand

#### 58 Citations

Bidimensionality, Map Graphs, and Grid Minors

- Computer Science, Mathematics
- ArXiv
- 2005

In this paper we extend the theory of bidimensionality to two families of graphs that do not exclude fixed minors: map graphs and power graphs. In both cases we prove a polynomial relation between… Expand

Bidimensional Parameters and Local Treewidth

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2004

It is proved that, for a large family of graph parameters called contraction-bidimensional, a minor-closed graph family has the parameter-treewidth property if $\mathcal{F}$ has bounded local treewidth. Expand

Linearity of grid minors in treewidth with applications through bidimensionality

- Mathematics, Computer Science
- Comb.
- 2008

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth,… Expand

A 2O(k)n algorithm for k-cycle in minor-closed graph families

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 2020

A randomized algorithm is presented that given a graph G ∈ C with n vertices, finds a simple cycle of size k in G (if exists) in 2 O ( k ) n time. Expand

Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs

- Mathematics, Computer Science
- JACM
- 2005

We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1). Our results apply to a broad family of graph problems, called… Expand

On the Maximum Independent Set Problem in Subclasses of Planar Graphs

- Computer Science, Mathematics
- J. Graph Algorithms Appl.
- 2010

By combining various techniques, it is shown that the maximum independent set problem is polynomially solvable in the class of S1,2,k-free planar graphs, generalizing several previously known results. Expand

From Tree-Width to Clique-Width: Excluding a Unit Interval Graph

- Mathematics, Computer Science
- ISAAC
- 2008

It is proved that the unit interval graphs constitute a minimal hereditary class of unbounded clique-width, and it is shown that list coloring is fixed parameter tractable in the class of unit intervals graphs. Expand

Vertex-Bipartition Method for Colouring Minor-Closed Classes of Graphs

- Computer Science, Mathematics
- Combinatorics, Probability and Computing
- 2010

It is proved that the values ρ() can be made arbitrarily large in the clique sum operation, and these results give rise to P-time approximation algorithms for colouring any graph in these classes within an error of at most 7 and n + 2 of its chromatic number, respectively. Expand

Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality

- Mathematics, Computer Science
- SODA '05
- 2005

We prove that any H-minor-free graph, for a fixed graph H, of treewidth ω has an Ω(ω) × Ω(ω) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth,… Expand

Layered Separators in Minor-Closed Families with Applications

- Mathematics
- 2013

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as ( p n) in… Expand

#### References

SHOWING 1-10 OF 16 REFERENCES

Diameter and Treewidth in Minor-Closed Graph Families

- 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

Local Tree-Width, Excluded Minors,
and Approximation Algorithms

- Mathematics, Computer Science
- Comb.
- 2003

A decomposition theorem for graphs with excluded minors says that such graphs can be decomposed into trees of graphs of almost bounded local tree-width, and it is shown that a number of combinatorial optimization problems have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor. Expand

Subgraph Isomorphism, log-Bounded Fragmentation and Graphs of (Locally) Bounded Treewidth

- Mathematics, Computer Science
- MFCS
- 2002

A new property for graphs (a generalization on bounded degree) is introduced and the known classes of inputs for which polynomial-time subgraph isomorphism algorithms are attainable are extended to handle graphs of locally bounded treewidth. Expand

Quickly Excluding a Planar Graph

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1994

A much better bound is proved on the tree-width of planar graphs with no minor isomorphic to a g × g grid and this is the best known bound. Expand

Approximation algorithms for classes of graphs excluding single-crossing graphs as minors

- Computer Science, Mathematics
- J. Comput. Syst. Sci.
- 2004

This paper demonstrates structural properties of larger classes of graphs and shows how to exploit the properties to derive polynomial-time algorithms for approximating treewidth and branchwidth and fixed-parameter algorithms for problems such as vertex cover, edge-dominating set, feedback vertex set, and others. 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

Highly Connected Sets and the Excluded Grid Theorem

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1999

A very simple obstruction to small tree-width is proposed, showing that a graph has small Tree-width if and only if it contains no large highly connected set of vertices. Expand

Deciding first-order properties of locally tree-decomposable structures

- Mathematics, Computer Science
- JACM
- 2001

It is shown that for each property φ of structures that is definable in first-order logic and for each locally tree-decomposable class C of structures, there is a linear time algorithm deciding whether a given structure A ∈ C hasproperty φ. Expand

Approximation algorithms for NP-complete problems on planar graphs

- 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

Graph Theory

- Computer Science
- 1997

Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results. Expand