In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé, and Watrigant [FOCS 2020] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced-$f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced-bandwidth, which implies and is stronger than bounded twin-width (reduced-maximum-degree).
We show that every proper minor-closed class has bounded reduced-bandwidth, which is qualitatively stronger than a result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced-bandwidth at most $466$ and twin-width at most $583$; moreover, the witnessing reduction sequence can be constructed in polynomial time. We show that $d$-powers of graphs in a proper minor-closed class have bounded reduced-bandwidth (irrespective of the degree of the vertices).
This is joint work with Édouard bonnet and David Wood.
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