How to delete categorically - Two pushout complement constructions

In category theory, most set-theoretic constructions—union, intersection, etc.—have direct categorical counterparts. But up to now, there is no direct construction of a deletion operation like the set-theoretic complement. In rule-based transformation systems, deletion of parts of a given object is one of the main tasks. In the double pushout approach to algebraic graph transformation, the construction of pushout complements is used in order to locally delete structures from graphs. But in general categories, even if they have pushouts, pushout complements do not necessarily exist or are unique. In this paper, two different constructions for pushout complements are given and compared. Both constructions are based on certain universal constructions in the sense of category theory. More specifically, one uses initial pushouts while the other one uses quasi-coproduct complements. These constructions are applied to examples in the categories of graphs and simple graphs.