Reachability, distance, and matching are some of the most fundamental graph problems that have been of particular interest in dynamic complexity theory in recent years [DKMSZ18, DMVZ18, DKMTVZ20]. Reachability can be maintained with first-order update formulas, or equivalently in DynFO in general graphs with n nodes [DKMSZ18], even under O(log n/loglog n) changes per step [DMVZ18]. In the context of how large the number of changes can be handled, it has recently been shown [DKMTVZ20] that under a polylogarithmic number of changes, reachability is in DynFOpar in planar, bounded treewidth, and related graph classes -- in fact in any graph where small non-zero circulation weights can be computed in NC. We continue this line of investigation and extend the meta-theorem for reachability to distance and bipartite maximum matching with the same bounds. These are amongst the most general classes of graphs known where we can maintain these problems deterministically without using a majority quantifier and even maintain witnesses. For the bipartite matching result, modifying the approach from [FGT], we convert the static non-zero circulation weights to dynamic matching-isolating weights. While reachability is in DynFOar under O(log n/loglog n) changes, no such bound is known for either distance or matching in any non-trivial class of graphs under non-constant changes. We show that, in the same classes of graphs as before, bipartite maximum matching is in DynFOar under O(log n/loglog n) changes per step. En route to showing this we prove that the rank of a matrix can be maintained in DynFOar, also under O(log n/loglog n) entry changes, improving upon the previous O(1) bound [DKMSZ18]. This implies similar extension for the non-uniform DynFO bound for maximum matching in general graphs and an alternate algorithm for maintaining reachability under O(log n/loglog n) changes [DMVZ18].

翻译：可见性、 距离和匹配是一些最根本的图形问题, 这些问题近年来对动态复杂理论特别感兴趣[ DKMSDZ18、 DMVZ18、 DKMTVZ20] 。 可用第一阶更新公式来维持, 或等同于Nn节的 DKMSZ18一般图形中的 DynFO 。 即使是在O(log n/log n) 每步变化 [DMVZ18] 下。 在如何处理大量变化的情况下, 最近显示 [DKMTVZ20] 在多logal- 动态复杂理论中 [DMVZ18、 DMMVVVZ20] 。 在多logal- 变异性变异的多数中, 在平面的平面变异变异变中, 最大变异性变异变异性在O- 变异性变异性变异性中, 最常变异性变异的变异性在我们所知道的图形中 。 在平面变异性变异性变异性变变异中, 最常的变异性变变变变变的变变变变变变, 以中, 我们的变异性变的变的变的变变变变变的变的变的变的变的变在的变的变的变的变变变的变,