A vertex subset of a (possibly infinite) graph is a prefiber (gated set) when from each exterior vertex all vertices of this subset are reachable by a geodesic (shortest path) which passes through one particular subset-vertex, called its projection on the prefiber. The graph is fiber-complemented (FC) if inverse projection-images of prefibers are always prefibers. All FC graphs may be constructed by iterated expansions of \(K_1\) by inclusion-minimal elementary graphs, and also by amalgamations of Cartesian products of such elementary graphs (a characterisation in general remains open). Median-, quasi-median-, pseudo-median-, weakly median- and bridged graphs are FC graphs, generated by particular (and known) families of elementary graphs. Any FC graph is isometrically embeddable in a Cartesian product of elementary graphs. Its prefibers generate a convexity structure and hence a topology on the vertex set, for which a full characterisation of compactness and compact Hausdorffness is obtained. This structural analysis allows to derive in a unified way (including infinite graphs) several other structural properties previously obtained for the different special subclasses of finite FC graphs.