  Bojanczyk and Pilipczuk showed in their celebrated article "Definability
equals recognizability for graphs of bounded treewidth" (LICS 2016) that
monadic second-order logic can define tree-decompositions in graphs of bounded
treewidth. This raises the question whether such decompositions can already be
defined in first-order logic (FO).
  We start by introducing the notion of tree-decompositions of bounded span,
which restricts the diameter of the subtree consisting of the bags containing a
same node of the structure. Having a bounded span is a natural property of
tree-decompositions when dealing with FO, since equality of nodes cannot in
general be recovered in FO when it doesn't hold. In particular, it encompasses
the notion of domino tree-decompositions.
  We show that path-decompositions of bounded span are not FO-continuous, in
the sense that there exist arbitrarily FO-similar graphs of bounded pathwidth
which do not possess FO-similar path-decompositions of bounded span. Then, we
show that tree-decompositions of bounded span are not FO-continuous either.
