By a graph, I mean a simple, undirected graph with no loops. A graph homomorphism $f : G \to H$ is a function from the vertexset of $G$ to the vertexset of $H$ such that if $u$ and $v$ are adjacent vertices in $G$, then $f(u)$ and $f(v)$ are adjacent vertices in $H$. A graph is said to be a core if all its endomorphisms are in fact automorphisms.
A vertex $v$ of a graph $G$ is said to be dismantlable if there is another vertex $w$ whose neighbourhood contains the neighbourhood of $v$. A graph with no dismantlable vertices is said to be stiff.
Every core is stiff. Indeed, if $v$ is a dismantlable vertex of $G$, so that there is some vertex $w \neq v$ with $N(w) \supseteq N(v)$, then the endomap $f : G \to G$ given by
$$f(u) = \begin{cases}
w, & \text{if } u = v; \\
u, &\text{if } u \neq v,
\end{cases}$$
is a graph homomorphism that is not surjective. Thus each core cannot have any dismantable vertices, and hence is stiff by definition.
My question is regarding the converse. Are there stiff graphs that are not cores?
