Obtaining a Bipartite Graph by Contracting Few Edges

@article{Heggernes2011ObtainingAB,
  title={Obtaining a Bipartite Graph by Contracting Few Edges},
  author={Pinar Heggernes and Pim van 't Hof and Daniel Lokshtanov and Christophe Paul},
  journal={SIAM J. Discret. Math.},
  year={2011},
  volume={27},
  pages={2143-2156}
}
The Bipartite Contraction problem takes as input an $n$-vertex graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. We show that Bipartite Contraction is fixed-parameter tractable when parameterized by $k$. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable… 

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References

SHOWING 1-10 OF 39 REFERENCES

Planarity Allowing Few Error Vertices in Linear Time

  • K. Kawarabayashi
  • Mathematics
    2009 50th Annual IEEE Symposium on Foundations of Computer Science
  • 2009
TLDR
The algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar in linear time, and extends several optimization results for planar graphs to k-apex graphs.

Edge-Contraction Problems

Odd cycle packing

TLDR
The integrality gap of the standard LP-relaxation of the odd cycle packing problem is Θ (√n), and it is proved that there is an O(m1/2)-approximation algorithm for the node- and arc- directed even cycle packing problems, which almost matches the hardness result.

Algorithm Engineering for Optimal Graph Bipartization

TLDR
This work examines exact algorithms for the NP-complete Graph Bipartization problem that asks for a minimum set of vertices to delete from a graph to make it bipartite and presents new algorithms and experimental results.

Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization

An Improved Algorithm for Finding Cycles Through Elements

We consider the following problem: Given k independent edges in G. Is there a polynomial time algorithm to decide whether or not G has a cycle through all of these edges ? If the answer is yes,

The computational complexity of graph contractions I: Polynomially solvable and NP‐complete cases

TLDR
Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that isknown to be NP-complete, thepattern graph H does not have a dominating vertices.

A simpler algorithm and shorter proof for the graph minor decomposition

TLDR
A simplified algorithm for finding the decomposition based on a new constructive proof of the decompose theorem for graphs excluding a fixed minor H, which runs in time O(n3), as does the original algorithm of Robertson and Seymour.

Clustering with local restrictions

A shorter proof of the graph minor algorithm: the unique linkage theorem

TLDR
This paper provides a new and much simpler proof of the correctness of the Graph Minor Algorithm and proves the "Unique Linkage Theorem" without using Graph Minors structure theorem.