Breakability of wall-free graphs without induced wall subdivisions

Prove that, for every positive integer t, there exists an integer d=d(t) such that every graph with no induced subgraph isomorphic to a line graph of a subdivision of the t-by-t wall and with no induced subgraph isomorphic to a subdivision of the t-by-t wall is d-breakable.

Background

A graph is d-breakable when every vertex-weight function admits a balanced separator contained in the closed neighborhood of fewer than d vertices. The paper proves this property for graphs excluding both the line graphs of subdivisions of a fixed wall and a fixed subdivided claw.

The conjecture proposes removing the subdivided-claw exclusion and replacing it with the condition that the graph itself has no induced subdivision of the wall. Establishing it would extend the paper’s bounded-core separator program to a substantially broader class of graphs.

References

It provides support for the following conjecture that was posed in and seems to be gaining popularity in the community: \begin{conjecture}\label{conj:domsep} For every positive integer $t$, there is an integer $d=d(t)$ such that every $\mathcal{L}_t$-free graph $G$ with no induced subgraph isomorphic to a subdivision of the $t \times t$-wall is $d$-breakable. \end{conjecture}

— Tree independence number V. Walls and claws  (2501.14658 - Chudnovsky et al., 24 Jan 2025) in Section 1, Conjecture 2 (labelled Conjecture~\ref{conj:domsep})