Absolute retracts in graphs

Abstract

If Y is a subgraph of X, any homomorphism r: X → Y satisfying r(y) = y for all y ε V(Y) is termed a retraction of X onto Y, and Y a retract of X. Let C denote a class of graphs, together with a class of homomorphisms; we investigate the class AR(C) of all absolute retracts of C, i.e., the class of all graphs Y ε C that are retracts of any supergraph X ε C for which the inclusion is a homomorphism in C.

We consider various classes C, e.g., the class P of all finite planar graphs and all homomorphisms, the class B of all bipartite graphs and metric homomorphisms, etc. In particular, we show that the Four Color Conjecture is equivalent with the statement AR(P) ≠ 0, and if the Four Color Conjecture is true we have a complete description of AR(P). We give a "good characterization" of AB(B), and obtain partial results for other interesting classes of graphs.