Definition 4.16 (2-lift). A 2-lift of a graph $G=(V,E)$ is a graph $Ĝ=(\widehat{V},Ê)$ with

• $x\in V$ $⟹$ $x_0,x_1\in \widehat{V}$.

• $xy\in E$ $⟹$ either $x_0{y}_0,x_1{y}_1\in Ê$ or $x_0{y}_1,x_1{y}_0\in Ê$.

(and no other vertices or edges).