Equations for higher Legendre transforms in terms of 1-irreducible vertices

For the functional Legendre transforms [2, 3] of arbitrary order equations are obtained in which 1-irreducible (and not simply connected) vertices are regarded as independent variables; their iterative solution is represented by skeleton graphs with 1-irreducible $n$-leg diagrams at their vertices. In this manner any $n$-leg vertex can be represented as the sum of a “bare” vertex and skeleton graphs of this type.