The combinatorial criterion for kinematic support

Let $\Gamma_{L^\ast}$ be a graph in the class of loop BCFW graphs $\mathbf{\Gamma}^{\mathtt{BCFW}}_{k,n;L^\ast}$, and let

$$(\mathcal{G}_0,\mathcal{G}_1,\dots,\mathcal{G}_T=\Gamma_{L^\ast})$$

be the corresponding branch of the loop BCFW recursion. Say that $\Gamma_{L^\ast}$ has kinematic support when it satisfies the realizability condition defined in the paper. Combinatorial criterion for kinematic support. The graph $\Gamma_{L^\ast}$ has kinematic support if and only if, for all faces $f^\ast,g^\ast$ and all $t=0,1,\dots,T$ for which they are faces of $\mathcal{G}_t$, the faces are $2_{\mathcal{N}}$-separated in $\mathcal{G}_t$ if and only if they are $2_{\mathcal{N}}$-separated in $\mathcal{G}_T=\Gamma_{L^\ast}$. The criterion is presented after an example showing that full $2$-separation alone is not sufficient for kinematic support. Its resolution status is not specified in the supplied text.