Unavoidable induced subgraphs in graphs with complete bipartite induced minors

TL;DR

This work investigates how large induced minors constrain the appearance of small induced subgraphs. By introducing structural types for how subgraphs interact with partitions and proving two main results—$K_{134,12}$ as an induced minor forces a short cycle or a theta, and $K_{3,4}$ as an induced minor forces a triangle or a theta—the authors draw connections to layered wheels and the limits of bounding tree-independence number via excluded grids or bipartite graphs. The results refine our understanding of induced-minor obstructions and provide a framework around 3-path configurations (theta/prism/pyramid) that extends to even-hole-free layered wheels. The findings have implications for the study of tree-independence number and related combinatorial structures in sparse and structured graph classes.

Abstract

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).

Figures (14)

• Figure 1: A $K_{3, 3}$ induced minor in a layered wheel
• Figure 2: The line graph of the graph obtained by subdividing each edge of $K_{3, 4}$
• Figure 3: A (triangle, theta)-free graph containing $K_{3, 3}$ as an induced minor
• Figure 4: A (theta, triangle)-free graph containing $K_{2, \ell}$ as an induced minor
• Figure 5: The different $3$-path configurations

Theorems & Definitions (31)

• Lemma 1.1: see DBLP:journals/jctb/DallardMS24
• Theorem 1.2: see DBLP:journals/jgt/SintiariT21
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: see DBLP:journals/jgt/SintiariT21
• Lemma 1.6
• Lemma 2.1
• proof
• Lemma 2.2