for the set of allowed poles, for CHY formula of $C_n^2$ , which we will denote as $P(C_n)$. Given the labelled tree, any of its non-trivial [*connected subgraph*]{} corresponds to a pole on the RHS of :

\[pc\] P(C\_n)={s\_[i\_1,i\_2,...,i\_m]{}| . } This rule for the poles is very intuitive, and it follows from the general analysis of pole structures of CHY formulas [@Cachazo:2013gna; @Baadsgaard:2015voa; @Cachazo:2015nwa; @Dolan:2013isa; @Broedel:2013tta] (for $C_n^2$ we only have simple poles, see [@Huang:2016zzb; @Cardona:2016gon] for discussions on higher-order poles). A connected subgraph with vertices $i_1, i_2, ...,i_m$ means that there are exactly $2(m-1)$ $\sigma_{i,j}$ with $i,j\in {i_1,...,i_m}$ in $C_n^2$, thus it will produce a pole $s_{i_1,i_2,...,i_m}$ according to the rule described in [@Baadsgaard:2015voa]. Note that we don’t have $n$ contained in any subgraph as any pole containing $n$ can be expressed by its complement.

Here we spell out some examples. For two-particle pole, $s_{i,j}\in P(C_n)$ iff is a edge in the labelled tree. For three particle pole, $s_{i,j,k}\in P(C_n)$ iff one of the graphs in figure \[sijk\] exists in the labelled tree.

For $m>3$, there are more topologies of subgraphs. For example, for $s_{i,j,k,l}$ there are two different topologies of subgraphs , see figure \[sijkl\].

With the 