Unavoidable induced subgraphs in graphs with complete bipartite induced minors
<|reference_start|>Unavoidable induced subgraphs in graphs with complete bipartite induced minors: We prove that if a graph contains the complete bipartite graph $K_{134, 12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a graph contains $K_{3, 4}$ as an induced minor, then it contains a triangle or a theta as an induced subgraph. Here, a \emph{theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = a \dots b$, $P_2 = a \dots b$, $P_3 = a \dots b$, each of length at least two, such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$.   A consequence is that excluding a grid and a complete bipartite graph as induced minors is not enough to guarantee a bounded tree-independence number, or even that the treewidth is bounded by a function of the size of the maximum clique, because the existence of graphs with large treewidth that contain no triangles or thetas as induced subgraphs is already known (the so-called layered wheels).<|reference_end|>
@article{chudnovsky2024unavoidable,
    title={Unavoidable induced subgraphs in graphs with complete bipartite induced
  minors},
    author={Maria Chudnovsky, Meike Hatzel, Tuukka Korhonen, Nicolas Trotignon and
  Sebastian Wiederrecht},
    journal={arXiv preprint arXiv:2405.01879},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01879},
    primaryClass={math.CO cs.DM}
}