Accessible pointed membership graphs

An accessible pointed graph consists of a set $X$ of nodes, a root $r\in X$, and a relation $R\subseteq X^2$. Draw an arrow $x\to y$ precisely when $yRx$. Accessibility means that for every $x\in X$ there are $n\in \omega$ and a function $p:n+1\to X$ with $p(0)=r$, $p(n)=x$, and $p(i+1)Rp(i)$ for $i<n$. The path of length zero reaches the root.

A decoration is a set function $d$ on $X$ such that $d(x)=\{d(y):yRx\}$ for each node. A well-founded graph means that $R$ has the minimal-element property, not an unqualified no-infinite-path characterization. Extensionality is not part of the graph definition. No axiom of anti-foundation is assumed.

Conventions and prerequisites: Well-founded and setlike relations.