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.

翻译：众所周知，图上的普通持久性计算比一般持久性更高效。近期研究表明，图上之字形持久性也展现出类似特性。受这些结果启发，我们重新审视图持久性，并针对过滤过程的局部更新提出高效算法，这类似于普通持久性中计算"葡萄园"的方法。我们证明：对于长度为$m$的过滤序列，(i) 普通图持久性中的交换操作（转置）可在$O(\log m)$时间内完成；(ii) 图上之字形持久性可在$O(m\log m)$时间内计算，这改进了近期一个$O(m\log^4n)$时间算法（假设过滤中所有图的并集大小$n$满足$n\in\Omega({m^\varepsilon})$，其中$0<\varepsilon<1$为任意固定值）；(iii) 图之字形持久性中0维的开闭、闭开及闭闭条可在$O(\log m)$时间内更新，而0维开开条与1维闭闭条可在$O(\sqrt{m}\,\log m)$时间内更新。