isWellDefined C
If $\{f_x \mid x\in X^T\}$ is a collection of Laurent polynomials in the character ring $\mathbb Z[T_0, \ldots, T_n]$ of the torus $T$ acting on a GKMVariety $X$, one per each torusfixed point, representing an element $C$ of $K_T^0(X^T)$, then $C$ is in the image of $K_T^0(X)$ under the injective restriction map $K_T^0(X)\to K_T^0(X^T)$ if and only if it satisfies the following "edge compatibility condition":
For each onedimensional $T$orbitclosure in $X$ with boundary points $x$ and $x'$, one has $$f_x \equiv f_{x'} \ \mod \ 1  T^{\lambda(x,x')}$$ where $\lambda(x,x')$ is the character of the action of $T$ on the onedimensional orbit. See [Corollary 5.12; VV03] or [Corollary A.5; RK03] for details.




A MomentGraph must be defined on the GKMVariety on which the KClass is a $K$class of.