# Dominating Sets and Local Treewidth

@inproceedings{Fomin2003DominatingSA, title={Dominating Sets and Local Treewidth}, author={F. Fomin and Dimitrios M. Thilikos}, booktitle={ESA}, year={2003} }

It is known that the treewidth of a planar graph with a dominating set of size d is \(O(\sqrt{d})\) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family \(\mathcal{F}\) has the domination-treewidth property if there is some function f(d) such that every graph \(G \in \mathcal{F}\) with dominating set of size…

## 11 Citations

Bidimensional Parameters and Local Treewidth

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2004

This paper examines the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far.

A refined search tree technique for Dominating Set on planar graphs

- Mathematics, Computer Science
- 2005

Refined Search Tree Technique for DOMINATING SET on Planar Graphs

- Mathematics, Computer ScienceMFCS
- 2001

A fixed parameter algorithm with running time O(8kn), where k is the size of the dominating set and n is the number of vertices in the graph is derived.

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

- Mathematics, Computer ScienceSODA '04
- 2004

A new framework for designing fixed-parameter algorithms with subexponential running time and a general approach for developing algorithms on H-minor-free graphs, based on structural results about Robertson & Seymour's graph-minors work are introduced.

Polynomial-time data reduction for dominating set

- Computer Science, MathematicsJACM
- 2004

It is proved that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy-to-implement reduction rules.

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

- Mathematics, Computer ScienceQuantum Inf. Process.
- 2017

Focusing on embedding into the hardware graph of commercially available quantum annealers, the minor set cover (MSC) of a known graph G is introduced: a subset of graph minors which contain any remaining minor of the graph as a subgraph.

Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

- Mathematics, Computer ScienceICALP
- 2001

Dominating sets in planar graphs: branch-width and exponential speed-up

- Computer ScienceSODA '03
- 2003

This paper shows how very deep min-max and duality theorems from Graph Minors can be used to obtain essential speed-up to many known algorithms on different domination problems.

SYSTEMATIC KERNELIZATION IN FPT ALGORITHM DESIGN

- Computer Science
- 2005

This thesis shows how the method of extremal structure can be applied to effectively solve several NP-complete problems, namely MAX CUT, MAX LEAF SPANNING TREE, NONBLOCKER, -STAR PACKing, EDGE-DISJOINT TRIANGLE PACKING, MAX INTERNAL SPANning TREE and MINIMUM MAXIMAL MATCHING.

Adiabatic quantum programming: minor embedding with hard faults

- Computer ScienceQuantum Inf. Process.
- 2014

Algorithms for embedding arbitrary instances of the adiabatic quantum optimization algorithm into a square lattice of specialized unit cells are presented and are shown to be more resilient to faulty fabrics than naive embedding approaches.

## References

SHOWING 1-10 OF 25 REFERENCES

Diameter and Treewidth in Minor-Closed Graph Families

- MathematicsAlgorithmica
- 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.

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

- MathematicsAlgorithmica
- 2002

An algorithm is presented that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) where c=4^ 6\sqrt 34 and k is the size of the face cover set.

Local Tree-Width, Excluded Minors,
and Approximation Algorithms

- MathematicsComb.
- 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.

Refined Search Tree Technique for DOMINATING SET on Planar Graphs

- Mathematics, Computer ScienceMFCS
- 2001

A fixed parameter algorithm with running time O(8kn), where k is the size of the dominating set and n is the number of vertices in the graph is derived.

Graph Minors. II. Algorithmic Aspects of Tree-Width

- Mathematics, Computer ScienceJ. Algorithms
- 1986

Dominating Sets in Planar Graphs

- Mathematics, Computer ScienceEur. J. Comb.
- 1996

It is proved that 1/4<e<1/3, and it is conjecture that e=1/4, and the upper bound proof yields a linear-time algorithm for finding an(n/3)-size dominating set.

Approximation algorithms for NP-complete problems on planar graphs

- Mathematics, Computer Science24th 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.

Improved Parameterized Algorithms for Planar Dominating Set

- Computer Science, MathematicsMFCS
- 2002

These algorithms induce a significant improvement over the previous best algorithm for the problem and can compute a dominating set of size bounded by k or report that no such set exists in time O(227?kn), where n is the number of vertices in G.

Deciding first-order properties of locally tree-decomposable structures

- Mathematics, Computer ScienceJACM
- 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.