Contractibility for digraphs and the fixed clique property

Homomorphism graphs are digraphs whose vertices are homomorphisms. A digraph is said to be contractible if the homomorphism graph consisting of vertices the self-mapping homomorphisms of the digraph is connected. In this paper, we show that the notion of contractible digraph extends and unifies various notions of dismantlable structures such as dismantlable graphs and dismantlable posets.