# 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…

## 44 Citations

### Reducing graph transversals via edge contractions

- MathematicsMFCS
- 2020

This work proves co-NP-hardness results under some assumptions on the graphs in ${\cal H}$, which implies that Contraction ($\pi$) is co- NP-hard even for fixed $k=d=1$ when $\pi$ is the size of a minimum feedback vertex set or an odd cycle transversal.

### On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges

- MathematicsIPEC
- 2013

Reducing the parameterized complexity of the contractibility problem for graph modification questions of the deletion variety shows that it is W[2]-hard to determine if at most k edges can be contracted to modify the given graph into a chordal graph.

### A faster FPT algorithm for Bipartite Contraction

- Computer Science, MathematicsInf. Process. Lett.
- 2013

### Contraction Blockers for Graphs with Forbidden Induced Paths

- MathematicsCIAC
- 2015

This work examines three graph parameters: the chromatic number, clique number and independence number, and shows that, when d is part of the input, this problem is polynomial-time solvable on P_4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs.

### Split Contraction

- MathematicsACM Trans. Comput. Theory
- 2019

This article examines an important family of graphs, namely, the family of split graphs, which in the context of edge contractions is proven to be significantly less obedient than one might expect and establishes the following conditional lower bound: SPLIT CONTRACTION cannot be solved in time 2o(ℓ2)⋅ nO(1), where ℓ is the vertex cover number of the input graph.

### On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

- MathematicsAPPROX-RANDOM
- 2020

This paper studies the problem of parameterized approximation of editing to a family of graphs by contracting edges by observing that the existing \textsf{W[2]-hardness} reduction can be adapted to show that there is no $F(k)$-\FPT-approximation algorithm for \textsc{Chordal Contraction}.

### Contracting Graphs to Paths and Trees

- Mathematics, Computer ScienceAlgorithmica
- 2012

It is shown that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly, which is surprising.

### Reducing the Vertex Cover Number via Edge Contractions

- Computer Science, MathematicsMFCS
- 2022

This article proves that it is NP - hard to decide whether an instance of Contraction( vc ) is a Yes -instance even when k = d, hence enhancing the understanding of the classical complexity of the problem.

### Parameterized Complexity of Two Edge Contraction Problems with Degree Constraints

- MathematicsIPEC
- 2013

Two graph modification problems where the goal is to obtain a graph whose vertices satisfy certain degree constraints are studied, and it is proved that both problems to be NP-complete for any fixed d ≥ 2.

## References

SHOWING 1-10 OF 39 REFERENCES

### Planarity Allowing Few Error Vertices in Linear Time

- Mathematics2009 50th Annual IEEE Symposium on Foundations of Computer Science
- 2009

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.

### Odd cycle packing

- MathematicsSTOC '10
- 2010

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

- Computer ScienceJ. Graph Algorithms Appl.
- 2005

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

- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 2006

### An Improved Algorithm for Finding Cycles Through Elements

- Computer Science, MathematicsIPCO
- 2008

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

- MathematicsNetworks
- 2008

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

- Mathematics, Computer ScienceSTOC '11
- 2011

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.

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

- Mathematics, Computer ScienceSTOC '10
- 2010

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.