Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors

We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

💡 Research Summary

This paper investigates the structural consequences of a graph containing a sufficiently large complete bipartite induced minor. The authors work within the broader program of relating induced subgraphs to treewidth, a central theme in modern graph theory. While the classical Robertson‑Seymour theory tells us that large treewidth forces the presence of a large grid (wall) minor, the induced‑subgraph analogue is far more intricate because many non‑basic obstructions appear, such as the Pohoata‑Davies graphs, occultations, and layered wheels. All these non‑basic examples share a common feature: they contain a large complete bipartite induced minor.

The paper introduces the notion of a constellation, a specially structured induced minor that models a complete bipartite graph K_{s,l} in a refined way. A constellation consists of a stable set S (the “stars”) and a collection L of vertex‑disjoint paths (the “tails”). Each vertex of S is adjacent to at least one vertex in every path of L. Consequently, any constellation contains an induced K_{s,l} minor and has treewidth at least min{s,l}. The authors prove Theorem 1.3: if a graph G has an induced K_{t,t} minor for sufficiently large t (depending on parameters r, s, ℓ), then either (a) G contains an induced wall W_{r×r}, or (b) G contains a large (s,ℓ)‑constellation as an induced subgraph. This result bridges the gap between large complete bipartite induced minors and the classical wall obstruction.

The second major contribution is a fine‑grained classification of the unavoidable induced subgraphs that can appear inside a large constellation. Two families are identified:

1. Interrupted constellations – there exists a linear order ≺ on S such that for any three vertices x≺y≺z, every S‑route (a path whose ends lie in S and whose interior lies in a single tail) between x and y meets a neighbor of z. This property exactly captures the previously studied “occultations”.

2. q‑zigzagged constellations – again with a linear order on S, for any pair x≺y, the number of intermediate vertices z (x≺z≺y) that avoid the S‑route between x and y is bounded by a constant q. When q = 2t, the authors speak of “2t‑zigzagged” constellations. This family generalises the Pohoata‑Davies graphs (also called “arrays”) but is strictly larger, as demonstrated by a constructed “zigzag sequence” Z_n that yields a 1‑zigzagged constellation with no large aligned sub‑constellation.

Theorem 2.1 formalises these ideas. For any prescribed parameters (d, ℓ, ℓ′, r, s, s′, t) there exist thresholds f₂.₁, g₂.₁ such that every (f₂.₁, g₂.₁)‑constellation either (a) contains a K_{r,r} or a proper subdivision of K_{2t+2}, or (b) contains a d‑ample interrupted (s,ℓ)‑constellation that “sits in” it, or (c) contains a d‑ample 2t‑zigzagged (s′,ℓ′)‑constellation that sits in it. The “sits in” relation preserves the structural properties (ample, interrupted, zigzagged) from the larger constellation to the smaller one.

A known lemma (Lemma 2.2, due to Aboulker et al.) is invoked to translate large wall induced minors into either a smaller wall or its line graph, completing the bridge between wall‑type and line‑graph‑type obstructions.

Finally, Theorem 2.³ combines Theorem 1.3, Theorem 2.1, and Lemma 2.2 to obtain a comprehensive trichotomy for any graph G that contains a sufficiently large K_{f,g} induced minor (with f,g given by a function of the parameters d, ℓ, ℓ′, r, s, s′). Specifically, G must contain at least one of the following:

• a K_{r,r} subgraph, a subdivision of a wall W_{r×r}, or the line graph of such a subdivision;
• a d‑ample interrupted (s,ℓ)‑constellation;
• a d‑ample 2r²‑zigzagged (s′,ℓ′)‑constellation.

The paper proceeds to justify that the third outcome cannot be strengthened to the more restrictive Pohoata‑Davies “array” structure (Section 3), and that both interrupted and zigzagged constellations are indeed unavoidable (Section 4). Sections 5‑8 develop the technical machinery needed to prove Theorem 2.1, handling the reduction to d‑ample constellations, constructing mixed intermediate structures, and finally separating the mixed case into the two clean outcomes. Section 9 completes the proof of Theorem 1.3, thereby establishing the full result.

In summary, the authors deliver a deep structural theorem: large complete bipartite induced minors force a graph either to contain a large wall (or its line graph) or to exhibit one of two highly organized constellation patterns. This advances our understanding of how induced minors dictate treewidth and provides a unified framework that captures all known non‑basic obstructions. The techniques introduced—particularly the constellation model and the notions of interruption and zigzagging—are likely to become standard tools in future investigations of induced‑minor theory, algorithmic graph decomposition, and the classification of graph classes with bounded treewidth.