On the nonexistence of FO-continuous path and tree-decompositions

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.