A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and ($s,t$)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an $m$-edge graph, there exists no dynamic algorithms with both $O(m^{1/2 - \epsilon})$ update time and $ O(m^{1 -\epsilon})$ query time, for any small $\epsilon > 0$. Note that for ($s,t$)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and $O(m)$ query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.

翻译：动态图算法是一种数据结构，它在支持边插入和删除等图修改操作的同时，回答关于当前图属性的查询。先前的研究已证明一般动态图存在强条件性下界，但在实际应用中出现的图族通常具有现有下界构造所不具备的结构性质。我们研究了三种普遍存在的特定图族，即常度数图、幂律图和扩展图，并给出了针对它们的首个条件性下界。结果表明，即使仅关注这些图类中的一种，任何用于基本图问题（如距离计算或近似、最大匹配）的算法都无法同时实现亚多项式更新时间和查询时间。例如，我们证明常度数图、幂律图或扩展图中最大匹配和(s,t)-距离的下界与一般图相同。具体而言，在具有m条边的图中，不存在更新时间为O(m^{1/2 - ε})且查询时间为O(m^{1 - ε})的动态算法，其中对任意小ε > 0均成立。值得注意的是，对于(s,t)-距离问题，平凡动态算法可实现常数更新时间和O(m)查询时间的几乎匹配上界。我们针对其他图族及其他基本问题（如最密子图检测和完美匹配）也证明了类似下界。