EPS is a notoriously large format, and there's no point using it for this because ListPlot3Ds can't be exported as vector graphics anyways.

Actually this is not true: for example, evaluating the following in version 10.2 produces a EPS file of size 66 Kb without any raster:

pl = ListPlot3D[{{1, 1, 1, 1}, {1, 2, 1, 2}, {1, 1, 3, 1}, {1, 2, 1, 4}}, Mesh -> All];
Export["pl.eps", pl]

I do not think 66 Kb EPS can be considered as a large file, especially If we take into consideration that Mathematica's EPS Export does not apply any compression which EPS format supports. As a comparison Exporting into PNG with good (but not excellent) resolution 300 dpi produces a file of size 136 Kb:

Export["pl.png", pl, ImageResolution -> 300]

In version 10 we face the fact that vector Exporting of 3D plots into PDF is disabled by default and even explicit setting of option AllowRasterization -> False does not work. I hope that it is a bug which will be fixed.

EPS is a notoriously large format, and there's no point using it for this because ListPlot3Ds can't be exported as vector graphics anyways. Try a raster format like Export["G1.png", Graph1] or Export["G1.jpg", Graph1].