First-order logic ($$FO$$) can express many algorithmic problems on graphs,  such as the independent set and dominating set problem parameterized by solution size. On the other hand, $$FO$$ cannot express the very simple algorithmic question whether two vertices are connected. We enrich $$FO$$ with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates $$conn_k(x,y,z_1,\ldots, z_k)$$that hold true in a graph if there exists a path between (the valuations of) $$x$$ and $$y$$ after (the valuations of) $$z_1,\ldots, z_k$$ have been deleted, we obtain separator logic. We thereby obtain a logic that can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes.