Twin-width of subdivisions of multigraphs
Abstract
For each d≤3, we construct a finite set Fd of multigraphs such that for each graph H of girth at least 5 obtained from a multigraph G by subdividing each edge at least two times, H has twin-width at most d if and only if G has no minor in Fd. This answers a question of Berg\'e, Bonnet, and D\'epr\'es asking for the structure of graphs G such that each long subdivision of G has twin-width 4. As a corollary, we show that the 7×7 grid has twin-width 4, which answers a question of Schidler and Szeider.
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