help of labelled tree, the relation of poles are also intuitive. Two poles [*i.e.*]{} two connected subgraphs , are compatible iff the particle set of one is a subset of that of the other, see figure \[compatible subgraph\] (1), or they have no intersection,see figure \[compatible subgraph\] (2). Two incompatible poles, see figure \[incompatible subgraph\] can’t both appear in a Feynman diagram.

We say a set of poles are compatible iff any two of them are compatible.

### Feynman diagrams from poles {#feynman-diagrams-from-poles .unnumbered}

Any $n-3$ compatible poles from $P(C_n)$ should correspond to a Feynman diagram on RHS of . In this subsubsection, we provide a clever way to obtain all Feynman diagrams from those of lowers points recursively, which is much more efficient than to find all $n-3$ compatible pole sets by brute force in higher points.

The starting point is that any cubic Feynman diagram has two parts whose external particles are $I,{\bar I}$ respectively which join a vertex with $n$, in the way shown in figure \[factozation2\]

According to , $s_I,s_{\bar I}$ corresponding to two connected subgraphs which together make up the whole labelled tree up to a link edge, see figure \[factozation\].

Reversely, each edge of 