Kernel of a digraph (directed graph) is a subset K of its nodes which is a) independet (no edges between nodes in K) and b) dominated by every outside node (having at least one edge to some node in K). (*) Every propositional theory T can be represented as a digraph G and every digraph G as a theory T in such a way that the models of T and kernels of G are in a one-to-one correspondence (and can be constructed from each other). In the finite case of (*), I briefly suggest applications for the algorithm design (transfer of techniques between SAT and kernel-search) as well as for a simple (purely propositional) yet adequate analysis of truth-referential paradoxes. (*) holds also for infinitary propositional logic, relating it to digraphs where nodes may have infinite out-degree. I sketch arguments for a series of equivalences (over $\operatorname{RCA}_0$) between (i) versions of the axiom of choice and (ii) kernel existence in various classes of digraphs.