Bounded-core separators for wall-free graphs

Establish that for every positive integer t there exists an integer d=d(t) such that every graph with no induced 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, meaning that every vertex weight function admits a balanced separator contained in the closed neighborhood of fewer than d vertices.

Background

A graph is d-breakable when every weight function has a balanced separator whose closed neighborhood is controlled by a core of fewer than d vertices. The paper proves a related result for graphs that are free of the line graphs of subdivisions of the t-by-t wall and free of the subdivided claw S_{t,t,t}.

The conjecture removes the subdivided-claw exclusion and instead explicitly requires that the graph have no induced subgraph isomorphic to a subdivision of the t-by-t wall. It is presented as a broader structural statement that could support further progress on polylogarithmic tree independence bounds.

References

\Cref{conj:qptas} is actually known to be implied by a~structural conjecture of~Gartland and Lokshtanov.

\begin{conjecture}[Gartland--Lokshtanov] For every planar graph $H$, there is a~constant $k := k(H)$ such that every $n$-vertex $H$-induced-minor-free graph $G$ admits a~set $X \subseteq V(G)$ of size at~most~$k$ such that the closed neighborhood of~$X$ is a~balanced separator of $G$. \end{conjecture}

— Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor  (2609.11285 - Bonnet et al., 10 Sep 2026) in Conjecture 2, Section 1, Introduction (subparagraph “Future work”)

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.

— Tree independence number V. Walls and claws  (2501.14658 - Chudnovsky et al., 24 Jan 2025) in Conjecture 2, Section 1 (Introduction)

It provides support for the following conjecture that was posed in and seems to be gaining popularity in the community: 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.

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