Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ 1\}\,\ E_e\ $ where

$$E_e\:=\ \{\{(a\ s)\,\ (b\ t)\}\ :\ \{a\ b\}\in V\ \ and\ \ t=e(\{a\ b\})\cdot s\}$$

• Now by looking at the $2$-lift pattern can one say if the lifted graph is connected or not?