  It is well known that ordinary persistence on graphs can be computed more
efficiently than the general persistence. Recently, it has been shown that
zigzag persistence on graphs also exhibits similar behavior. Motivated by these
results, we revisit graph persistence and propose efficient algorithms
especially for local updates on filtrations, similar to what is done in
ordinary persistence for computing the vineyard. We show that, for a filtration
of length $m$, (i) switches (transpositions) in ordinary graph persistence can
be done in $O(\log m)$ time; (ii) zigzag persistence on graphs can be computed
in $O(m\log m)$ time, which improves a recent $O(m\log^4n)$ time algorithm
assuming $n$, the size of the union of all graphs in the filtration, satisfies
$n\in\Omega({m^\varepsilon})$ for any fixed $0<\varepsilon<1$; (iii)
open-closed, closed-open, and closed-closed bars in dimension $0$ for graph
zigzag persistence can be updated in $O(\log m)$ time, whereas the open-open
bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in
$O(\sqrt{m}\,\log m)$ time.
