# Odd holes in bull-free graphs

@article{Chudnovsky2018OddHI, title={Odd holes in bull-free graphs}, author={M. Chudnovsky and Vaidy Sivaraman}, journal={ArXiv}, year={2018}, volume={abs/1704.04262} }

The complexity of testing whether a graph contains an induced odd cycle of length at least five is currently unknown. In this paper we show that this can be done in polynomial time if the input graph has no induced subgraph isomorphic to the bull (a triangle with two disjoint pendant edges).

## 8 Citations

Perfect divisibility and 2‐divisibility

- MathematicsJ. Graph Theory
- 2019

It is proved that if a graph is bull-free and either odd-hole-free or $P_5-free, then it is perfectly divisible.

FINDING LARGE H-COLORABLE SUBGRAPHS

- Mathematics
- 2020

We study the Max Partial H-Coloring problem: given a graph G, find the 5 largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern 6 graph without loops. Note that…

Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

- MathematicsSTACS
- 2022

The complexity to the time required to perform a poly-logarithmic number of multiplications of n2 × n2 Boolean matrices is reduced, leading to a largely improved O(n4.75)-time algorithm.

Three-in-a-tree in near linear time

- Computer ScienceSTOC
- 2020

The three-in-a-tree problem is solved in O(mlog2 n) time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes.

Finding large $H$-colorable subgraphs in hereditary graph classes

- MathematicsESA
- 2020

It is proved that for every fixed pattern graph H, the problem of finding the largest induced subgraph of G that admits a homomorphism into H can be solved, and that even a restricted variant of \textsc{Max Partial $H$-Coloring} is $\mathsf{NP}$-hard in the considered subclasses of $P_5$-free graphs, if the authors allow loops on $H$.

Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

- MathematicsSIAM J. Discret. Math.
- 2021

## References

SHOWING 1-7 OF 7 REFERENCES

Recognizing Berge Graphs

- MathematicsComb.
- 2005

This paper gives an algorithm to test if a graph G is Berge, with running time O(|V (G)|9), independent of the recent proof of the strong perfect graph conjecture.

The structure of bull-free graphs I - Three-edge-paths with centers and anticenters

- MathematicsJ. Comb. Theory, Ser. B
- 2012

Linear-time modular decomposition and efficient transitive orientation of comparability graphs

- MathematicsSODA '94
- 1994

The lirst linear-time algorithm for modular decomposition is given, and a new bound of 0 (ri +m logn) on transitive orientation and the problem of recognizing permutation graphs and two-dimensional partial orders is solved.

The strong perfect graph theorem

- Mathematics100 Years of Math Milestones
- 2019

In 1960 Berge came up with the concept of perfect graphs, and in doing so, conjectured some characteristics about them. A perfect graph is a graph in which the chromatic number of every induced…

The structure of bull - free graphs I — Three - edge - paths with centers and anti - centers , Journal of Combinatorial Theory

- 2012

By [3] we can find a homogeneous set in time O(|V (G)|). By 1.9 steps 2(a) takes time O

- Complexity analysis: Clearly step 1 takes time O Since |V (G 1 (X))| + |V (G 2 (X))| = |V (G)| + 1, it follows that the recursion of step 2(b) takes time O(|V (G)| 5 ). Consequently the algorithm runs in time O(|V (G)| 5 ), as claimed. ✷ References

Test if G contains C 5 by enumerating all 5-tuples. If yes, stop and output: " G contains an odd hole

- Test if G contains C 5 by enumerating all 5-tuples. If yes, stop and output: " G contains an odd hole