Polylogarithmic tree independence for graphs excluding induced-minor walls and bicliques

Establish, for every positive integer t, 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 main theorem obtains a fourth-power logarithmic bound on tree independence number when the graph excludes K_{t,t}, a fixed subdivided claw, and line graphs of subdivisions of a fixed wall. The conjecture seeks a general polylogarithmic bound under the more structural hypothesis of excluding K_{t,t} and the wall as induced minors.

The authors identify their bounded-core separator theorem, together with the layered-set methods developed in the paper, as promising steps toward this broader conjecture. The statement remains unresolved in the supplied paper.

References

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: \begin{conjecture}\label{conj:smalltreealph} 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 $\logd n$. \end{conjecture}

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