# Treewidth computation and extremal combinatorics

@article{Fomin2008TreewidthCA, title={Treewidth computation and extremal combinatorics}, author={F. Fomin and Yngve Villanger}, journal={Combinatorica}, year={2008}, volume={32}, pages={289-308} }

AbstractFor a given graph G and integers b,f ≥0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most $n\left( {_b^{b + f} } \right)$ such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide…

## 69 Citations

FPT Algorithms to Compute the Elimination Distance to Bipartite Graphs and More

- MathematicsWG
- 2021

This work presents non-uniform fixed-parameter tractable algorithms for testing whether the H-elimination distance or H-treewidth of a graph is at most k, and provides such algorithms for all graph classes H defined by a finite set of forbidden induced subgraphs.

Finding Induced Subgraphs via Minimal Triangulations

- MathematicsSTACS
- 2010

It is shown that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potentialmaximal cliques times O(n^{O(t)}) to find a maximum induced subgraph of treewidth t in G and for a given graph F to decide if G contains an induced sub graph isomorphic to F.

Homomorphisms are a good basis for counting small subgraphs

- MathematicsSTOC
- 2017

Graph motif parameters are introduced, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs, and a general complexity dichotomy for evaluating graph motif parameters is proved, which allows us to recover known dichotomies for counting sub graphs, induced sub Graph, and homomorphisms in a uniform and simplified way.

Large Induced Subgraphs via Triangulations and CMSO

- MathematicsSIAM J. Comput.
- 2015

It is shown that all potential maximal cliques of $G$ can be enumerated in time and implies the existence of an exact exponential algorithm of running time ${\cal O}(1.7347^n)$ for many NP-hard problems related to finding maximum induced subgraphs with different properties.

Faster Parameterized Algorithms for Minimum Fill-in

- Computer Science, MathematicsAlgorithmica
- 2010

A new lemma is presented describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k, which improves the base of the exponential part of the best known parameterized algorithm time for this problem so far.

A Parameterized Algorithm for Chordal Sandwich

- MathematicsCIAC
- 2010

This paper gives an algorithm with running time $O(2^{k}n^{5})$ to solve the Chordal Sandwich problem, and shows that the problem becomes tractable when parameterized with a suitable natural measure on the set of admissible edges F.

Parameterized Complexity of Deletion to Scattered Graph Classes

- Mathematics, Computer ScienceIPEC
- 2020

This paper initiates a study of a natural variation of the problem of deletion to scattered graph classes where the authors need to delete at most k vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes.

Efficient Algorithms for the max k -vertex cover Problem

- Computer Science, MathematicsIFIP TCS
- 2012

It is proved that, there exists an exact algorithm for maxk-vertex cover with complexity bounded above by the maximum among ck and γτ, for some γ<2, where τ is the cardinality of a minimum vertex cover of G.

On restricted completions of chordal and trivially perfect graphs

- MathematicsSSRN Electronic Journal
- 2022

Let G be a graph having a vertex v such that H = G − v is a trivially perfect graph. We give a polynomial-time algorithm for the problem of deciding whether it is possible to add at most k edges to G…

## References

SHOWING 1-10 OF 55 REFERENCES

Exact Algorithms for Treewidth and Minimum Fill-In

- Computer Science, MathematicsSIAM J. Comput.
- 2008

It is shown that the treewidth and the minimum fill-in of an $n$-vertex graph can be computed in time $\mathcal{O}(1.8899^n)$ and the running time of the algorithms can be reduced to 1.4142 minutes.

Graph Minors. II. Algorithmic Aspects of Tree-Width

- Mathematics, Computer ScienceJ. Algorithms
- 1986

Complexity of finding embeddings in a k -tree

- Mathematics, Computer Science
- 1987

This work determines the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree and presents an algorithm with polynomially bounded (but exponential in k) worst case time complexity.

Counting Clique Trees and Computing Perfect Elimination Schemes in Parallel

- Computer ScienceInf. Process. Lett.
- 1989

Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In

- Mathematics, Computer ScienceICALP
- 2004

We show that for a graph G on n vertices its treewidth and minimum fill-in can be computed roughly in 1.9601 n time. Our result is based on a combinatorial proof that the number of minimal separators…

Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In

- Computer ScienceLATIN
- 2006

It is shown that the number of potential maximal cliques for an arbitrary graph G on n vertices is ${\mathcal O}^{*}$(1.8135n), and that all potential maximalCliques can be listed in ${\ mathcal O]^{*]$ (1.8899n) time.

Improved approximation algorithms for minimum-weight vertex separators

- Mathematics, Computer ScienceSTOC '05
- 2005

The algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces is developed, and an O(√log n) pseudo-approximation for finding balanced vertices in general graphs is exhibited.

Combinatorial Optimization on Graphs of Bounded Treewidth

- Computer ScienceComput. J.
- 2008

The concepts of treewidth and tree decompositions are introduced, and the technique with the Weighted Independent Set problem is illustrated, to survey some of the latest developments.