Absolute Reflexive Retracts and Absolute Bipartite Retracts

Abstract It is a well-known phenomenon in the study of graph retractions that most results about absolute retracts in the class of bipartite (irreflexive) graphs have analogues about absolute retracts in the class of reflexive graphs, and vice versa. In this paper we make some observations that make the connection explicit. We develop four natural transformations between reflexive graphs and bipartite graphs which preserve the property of being an absolute retract, and allow us to derive results about absolute reflexive retracts from similar results about absolute bipartite retracts and conversely. Then we introduce generic notions that specialize to the appropriate concepts in both cases. This paves the way to a unified view of both theories, leading to absolute retracts of general (i.e., partially reflexive) graphs.