Graphs with the adjacent vertex property, but not the fixed clique property

Consider the class of fi nite graphs G such that G does not have the fi xed clique property and every proper simplicial retract has the fixed clique property. Within this class, we characterize the graphs such that every simplicial automorphism has a vertex that is adjacent or equal to its image, and such that the complement of G has maximum degree 3 and a vertex of degree 2. It turns out that this property is close to an NP-complete problem with a rather beautiful embedding of 3SAT into combinatorics.