Polylogarithmic tree independence for induced-minor-free graphs

Prove that for every positive integer t there exists an integer d=d(t) such that every n-vertex graph with no induced minor isomorphic to K_{t,t} or to the t-by-t wall W_{t\times t} has tree independence number at most log^d n.

Background

The paper’s separator theorem for graphs excluding line graphs of wall subdivisions and subdivided claws, together with a previously established result and the layered-set method, is described as progress toward a more general tree-independence conjecture.

This conjecture concerns graphs excluding K_{t,t} and the t-by-t wall as induced minors, and seeks a polynomial-in-logarithm upper bound on tree independence number. Unlike the paper’s main theorem, it uses induced-minor exclusions and does not include the subdivided-claw restriction.

References

For every positive integer $t$, there is an integer $d=d(t)$ such that for every $n\ge 2$, every $n$-vertex graph with no induced minor isomorphic to $K_{t,t}$ or to $W_{t\times t}$ has tree independence number at most $\logd n$.

— Tree independence number V. Walls and claws  (2501.14658 - Chudnovsky et al., 24 Jan 2025) in Conjecture 3, Section 1 (Introduction)

In turn, Conjecture~\ref{conj:domsep}, together with Theorem~\ref{few big independent neighborhoods in big independent set} and the methods of Section~\ref{sec:layeredsets}, are promising steps toward the following: For every positive integer $t$, there is an integer $d=d(t)$ such that for every $n\ge 2$, every $n$-vertex graph with no induced minor isomorphic to $K_{t,t}$ or to $W_{t\times t}$ has tree independence number at most $\logd n$.

— Tree independence number V. Walls and claws  (2501.14658 - Chudnovsky et al., 24 Jan 2025) in Section 1, Introduction, Conjecture 3 (labelled conj:smalltreealph)