Extensions on 2-edge connected 3-regular up-embeddable graphs

It is known⁽⁵⁾ that an investigation of the up-embeddability of the 3-regular graphs shows a useful approach to that of the general graph. But as far, very few characterizations of the up-embeddability are known on the 3-regular graphs. LetG be a 2-edge connected 3-regular graph. We prove thatG is up-embeddable if and only ifG can be obtained from the graphs ϑ, orK ₄ by a series ofM- orN-extensions. Meanwhile, we also present a new structural characterization of such graphG provided thatG is up-embeddable.