On a class of isometric subgraphs of a graph

In a graphG, which has a loop at every vertex, a connected subgraphH=(V(H),E(H)) is a retract if, for anya, b ∈V(H) and for any pathsP, Q inG, both joininga tob, and satisfying |Q|≧ ≧|P|, thenP ⫅V(H) wheneverQ ⫅V(H). As such subgraphs can be described by a closure operator we are led to the investigation of the corresponding complete lattice of “closed” subgraphs. For example, in this complete lattice every element is the infimum of an irredundant family of infimum irreducible elements. The work presented here was supported in part by N.S.E.R.C. Operating Grant No. A4077.