# Computing Tree Decompositions with Small Independence Number

@article{Dallard2022ComputingTD, title={Computing Tree Decompositions with Small Independence Number}, author={Cl{\'e}ment Dallard and F. Fomin and Petr A. Golovach and Tuukka Korhonen and Martin Milanivc}, journal={ArXiv}, year={2022}, volume={abs/2207.09993} }

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP -hard graph problems, like maximum weight independent set, can be solved in time n O ( k ) if the input graph is given with a tree decomposition of independence number at most k . However, it was an open problem if tree-independence number could be…

## One Citation

### Tree decompositions with bounded independence number: beyond independent sets

- Mathematics, Computer Science
- 2022

We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more speciﬁcally, of graph classes with bounded tree-independence number.…

## References

SHOWING 1-10 OF 36 REFERENCES

### Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure

- MathematicsArXiv
- 2022

We continue the study of (tw , ω )-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of…

### Approximating clique-width and branch-width

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2006

### Treewidth versus clique number. II. Tree-independence number

- Mathematics
- 2021

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw , ω )-bounded. The family of (tw , ω…

### Graph Minors. II. Algorithmic Aspects of Tree-Width

- Mathematics, Computer ScienceJ. Algorithms
- 1986

### Almost polynomial factor inapproximability for parameterized k-clique

- Mathematics, Computer ScienceCCC
- 2022

This paper improves the inapproximability result to rule out every F(k) = k1/H( k) factor FPT-approximation algorithm for any increasing computable function H (for example H(K) = log* k).

### Treewidth versus clique number in graph classes with a forbidden structure

- MathematicsWG
- 2020

The results imply that the class of $1$-perfectly orientable graphs is $(tw,\omega)$-bounded, answering a question of Bresar, Hartinger, Kos and Milanic from 2018.

### On H-Topological Intersection Graphs

- MathematicsWG
- 2017

It is proved that it is NP-complete if $H$ contains the diamond graph as a minor and that the clique problem is APX-hard, and it is shown that both the k-clique and the list ofcoloring problems are solvable in FPT-time on $H-graphs.

### A ck n 5-Approximation Algorithm for Treewidth

- Computer Science, MathematicsSIAM J. Comput.
- 2016

This is the first algorithm providing a constant factor approximation for treewidth which runs in time single exponential in $k$ and linear in the input size and can be used to speed up many algorithms to work in time.