Fixed point results for multivalued contractions on a metric space with a graph

We consider multivalued maps defined on a complete metric space endowed with a directed graph. We establish fixed point results for maps, called weak GG-contractions, which send connected points into connected points and contract the length of paths. The fixed point sets are studied. The homotopical invariance property of having a fixed point is also established for a family of weak GG-contractions.