  In this paper, we first establish the very close link between stability of
graphs, a concept first introduced in \cite{Scapsalvi1} and studied most
notably by Surowski \cite{Surowski1}, \cite{Surowski2} and Wilson
\cite{Wilson01} and two-fold automorphisms. The concept of two-fold
isomorphisms, as far as we know, first appeared in literature in the form of
isotopies of digraphs \cite{zelinka4}, \cite{zelinka1}, \cite{zelinka2},
\cite{zelinka3} and later studied formally in \cite{lms1}, \cite{lms2} with a
greater emphasis on undirected graphs. We then turn our attention to the
stability of graphs which have every edge on a triangle, but with the fresh
outlook provided by TF-automorphisms. Amongst such graphs are strongly regular
graphs with certain parameters. The advantages of this fresh outlook are
highlighted when we ultimately present a method of constructing and generating
unstable graphs with large diameter having every edge lying on a triangle. This
was a rather surprising outcome.
