The Laplacian measure of a function in $G$ is the weak limit of linear combinations of the Laplacian measures of your elementary functions, whose support are "tripods", that is, unions of three rays emanating from a vertex.
In particular, the support of the Laplacian measure of a function in $G$ must be a union of tripods. The example by Alexandre Eremenko has a Laplacian whose support cannot be expressed as a union of tripods.