  The stability result, as stated, is incorrect. In particular, the
1-dimensional extended persistence diagrams of a finely-triangulated simplicial
complex X equipped with a continuous real-valued function f, and its
one-skeleton graph G (also equipped with f), need not be close. To take an
example, let X be a finely-triangulated disc of radius r and let f be the
distance-to -the-boundary function, which increases radially from the boundary
circle of X and is maximized at the center. The extended persistence diagram of
(X,f) contains a point in 1-dimensional persistence of the form (0,r), whereas
the 1-dimensional extended persistence diagram of its one-skeleton only has
points near the diagonal.
