  Graph rules for the linked cluster expansion of the Legendre effective action
$\Gamma[\phi]$ are derived and proven in $D\geq 2$ Euclidean dimensions. A key
aspect is the weight assigned to articulation vertices which is itself shown to
be computable from labeled tree graphs. The hopping interaction is allowed to
be long ranged and scale dependent, thereby producing an in principle exact
solution of $\Gamma[\phi]$'s functional renormalization group equation.
