Absolute Retracts and Varieties of Reflexive Graphs

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Abstract
For a graph G, let V(G) denote its vertex set and E(G) its edge set. Here we shall only consider reflexive graphs, that is graphs in which every vertex is adjacent to itself. These adjacencies, i.e., the loops, will not be depicted in the figures, although we always assume them present. For graphs G and H, an edge-preserving map (or homomorphism) of G to H is a mapping of V(G) to V(H) such that f(g) is adjacent to f(g′) in H whenever g is adjacent to g′ in G. Because our graphs are reflexive, an edge-preserving map can identify adjacent vertices, i.e., possibly f(g) = f(g′) for some g adjacent to g′, cf. Figure 1(a).

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