A hereditary graph class has bounded shrub-depth if and only if it forbids flipped half-graphs and flipped unions of long paths, which happens exactly when MSO is no more expressive than FO.
Twin-width iv: Ordered graphs and matrices
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Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO
A hereditary graph class has bounded shrub-depth if and only if it forbids flipped half-graphs and flipped unions of long paths, which happens exactly when MSO is no more expressive than FO.