  A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also
called the canonical double cover of $X$) has automorphisms that do not come
from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable
if it is unstable, connected, non-bipartite, and distinct vertices have
distinct sets of neighbours. In this paper, we prove two sufficient conditions
for stability of graphs in which every edge lies on a triangle, revising an
incorrect claim of Surowski and filling in some gaps in the proof of another
one. We also consider triangle-free graphs, and prove that there are no
non-trivially unstable triangle-free graphs of diameter 2. An interesting
construction of non-trivially unstable graphs is given and several open
problems are posed.
