Fiber-complemented graphs II. Retractions and endomorphisms

AbstractA fiber-complemented graph is a graph for which the inverse image of every prefiber (or gated set) by any projection map onto a prefiber is a prefiber. In this paper, we continue the study of these graphs and establish a retraction theorem and fixed point properties for endomorphisms. Adding the notion of mooring (these are particular retractions of a graph onto its prefibers) to the tools introduced in Part I of this work (Discrete Math. 226 (2001) 107), we show that a fiber-complemented graph whose elementary prefibers induce moorable graphs is a retract of a Cartesian product of elementary moorable graphs. Then we deduce that under some conditions of compacticity, the elements of every commuting family of endomorphisms of a moorable pre-median graph strictly stabilize a nonempty finite pre-median subgraph (pre-median graphs are particular instances of weakly modular graphs which are fiber-complemented). These results give generalizations of analogous properties related to median graphs, quasi-median graphs, pseudo-median graphs and weakly median graphs