  A group $G$ splits over a subgroup $C$ if $G$ is either a free product with
amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A
\underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite
groups which admit cubic Cayley graphs of connectivity two in terms of
splittings over a subgroup.
