Unavoidable induced subgraphs in graphs with complete bipartite induced minors
arxiv(2024)
摘要
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 theta is a graph made of
three internally vertex-disjoint chordless paths P_1 = a … b, P_2 = a
… b, P_3 = a … 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).
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