It is proved definability in FO+IFP of a global linear ordering on vertices of strongly extensional (S E ) finitely branching graphs. In the case of finite S E graphs this also holds for FO+LFP. This gives capturing results for PTIME computability on the latter class of graphs by FO+LFP and FO+IFP, and also on the corresponding anti-founded universe HFA of hereditarily finite sets by a language Δ of a bounded set theory BSTA. Oracle PTIME computability over HFA is also captured by an appropriate extension of the language Δ by predicate variables and a bounded ϵ-recursion schema. It is also characterized the type of corresponding linear ordering on the universe HFA and on its natural extension HFA^∞ consisting of hereditarily finite anti-founded sets with possibly infinite (unlike HFA) transitive closures.