关于FO-连续路的不存在性与树分解
On the nonexistence of FO-continuous path and tree-decompositions
摘要：
Bojanczyk和Pilipczuk在他们的著名文章“Definability equals recognizability for graphs of bounded treewide”（LICS 2016）中表明，一元二阶逻辑可以在有界树宽的图中定义树分解。这就提出了一个问题，即这种分解是否已经可以在一阶逻辑（FO）中定义。我们首先引入有界跨度的树分解的概念，它限制了由包含结构的相同节点的袋组成的子树的直径。在处理FO时，具有有界跨度是树分解的自然属性，因为当FO不成立时，节点的相等通常不能在FO中恢复。特别是，它包含了多米诺树分解的概念。我们证明了有界跨度的路径分解不是FO-连续的，在这个意义上说，存在任意的有界路径宽度的FO-相似图不具有有界跨度的FO-相似路径分解。然后，我们证明了有界跨度的树分解也不是FO-连续的。
英文摘要：
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.
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