# Dominating set is fixed parameter tractable in claw-free graphs

@article{Cygan2010DominatingSI, title={Dominating set is fixed parameter tractable in claw-free graphs}, author={Marek Cygan and Geevarghese Philip and Marcin Pilipczuk and Michal Pilipczuk and Jakub Onufry Wojtaszczyk}, journal={Theor. Comput. Sci.}, year={2010}, volume={412}, pages={6982-7000} }

## 24 Citations

### Domination When the Stars Are Out

- MathematicsICALP
- 2011

It is shown that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel.

### Linear-time Algorithms for Eliminating Claws in Graphs

- Mathematics, Computer ScienceCOCOON
- 2020

It is shown that the problem of determining the distance of a given graph to a claw-free graph can be solved in linear time by a simpler algorithm on forests, and the exact values for full $k$-ary trees are determined.

### Parameterized Domination in Circle Graphs

- Mathematics, Computer ScienceTheory of Computing Systems
- 2013

It turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs, and the FPT algorithm runs in subexponential time.

### Parameterized complexity in multiple-interval graphs: Domination, partition, separation, irredundancy

- MathematicsTheor. Comput. Sci.
- 2012

### Connected $k$-partition of $k$-connected graphs and $c$-claw-free graphs

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2021

This work shows how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each wi is greater than the weight of the heaviest vertex.

### Online Domination: The Value of Getting to Know All your Neighbors

- Computer ScienceMFCS
- 2021

The results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood, whereas knowing the neighbors allows us to improve it to 2-competitiveness.

### Improved FPT algorithms for weighted independent set in bull-free graphs

- Mathematics, Computer ScienceDiscret. Math.
- 2014

### Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

- Mathematics, Computer ScienceAlgorithmica
- 2016

It is proved that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize, and a non-trivial sparsification algorithm is exhibited for d-Not-All-Equal-SAT.

### How to Eliminate a Graph

- Mathematics, Computer ScienceWG
- 2012

The parameterized complexity of the Elimination problem is studied and it is shown that Elimination is W[1]-hard when parameterized by |V(H)|, even if both input graphs are split graphs, and W[2]- hard when parameterize by |G(G)|−|V( H)|,even if H is a complete graph.

### Modifying a Graph Using Vertex Elimination

- Mathematics, Computer ScienceAlgorithmica
- 2013

The computational complexity of the problem depends on which graph is assumed to be a tree: it is shown that Elimination can be solved in polynomial time when H is a tree, whereas it remains NP-complete when G is a Tree.

## 41 References

### Domination When the Stars Are Out

- MathematicsICALP
- 2011

It is shown that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel.

### Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

- MathematicsAlgorithmica
- 2008

This paper gives a kO(dk)n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices, which proves that the dominating set problem is fixed-parameter tractable for degenerated graphs.

### On maximal independent sets of vertices in claw-free graphs

- MathematicsJ. Comb. Theory, Ser. B
- 1980

### Domination Problems in Nowhere-Dense Classes

- MathematicsFSTTCS
- 2009

It is shown that the distance-$d$ dominating-set problem, also known as the $(k,d)$-centres problem, is fixed-parameter tractable on any class that is nowhere dense and closed under induced subgraphs.

### Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels

- Mathematics, Computer ScienceESA
- 2009

We show that the k -Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K i,j as a subgraph, for any fixed i,j ≥ 1. This…

### Deciding First-Order Properties for Sparse Graphs

- Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

An almost linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion and a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion for testing existential properties or the existence of short paths between prescribed vertices.

### Fixed Parameter Tractability and Completeness

- MathematicsComplexity Theory: Current Research
- 1992

It is shown that if Dominating Set is fixed-parameter tractable, then so are a variety of parameterized problems, such as Independent Set, and that for this problem, and for the problem of determining whether a graph has k disjoint cycles, it may take c = 1.

### Optimization problems in multiple-interval graphs

- MathematicsSODA '07
- 2007

It is proved that MAXIMUM CLIQUE is NP-complete for the case of t = 3, and a (t2--t+1)/2-approximation algorithm for the problem is provided, using recent bounds proven for the so-called transversal number of t-interval families.