Separator logic and star-free expressions for graphs

We describe two formalisms for defining graph languages, and prove that they are equivalent: 1. Separator logic. This is first-order logic on graphs which is allowed to use the edge relation, and for every $n \in \{0,1,\ldots \}$ a relation of arity $n+2$ which says that "vertex $s$ can be connected to vertex $t$ by a path that avoids vertices $v_1,\ldots,v_n$". 2. Star-free graph expressions. These are expressions that describe graphs with distinguished vertices called ports, and which are built from finite languages via Boolean combinations and the operations on graphs with ports used to construct tree decompositions. Furthermore, we prove a variant of Sch\"utzenberger's theorem (about star-free languages being those recognized by a periodic monoids) for graphs of bounded pathwidth. A corollary is that, given $k$ and a graph language represented by an \mso formula, one can decide if the language can be defined in either of two equivalent formalisms on graphs of pathwidth at most $k$.