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). 
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8 Citations
Background Citations
3
Methods Citations
1

8 Citations

Citation Type

Citation Type

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TLDR
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TLDR
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

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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