Temporal graphs are graphs where the presence or properties of their vertices and edges change over time. When time is discrete, a temporal graph can be defined as a sequence of static graphs over a discrete time span, called lifetime, or as a single graph where each edge is associated with a specific set of time instants where the edge is alive. For static graphs, Courcelle's Theorem asserts that any graph problem expressible in monadic second-order logic can be solved in linear time on graphs of bounded tree-width. We propose the first adaptation of Courcelle's Theorem for monadic second-order logic on temporal graphs that does not explicitly rely on the lifetime as a parameter. We then introduce the notion of derivative over a sliding time window of a chosen size, and define the tree-width and twin-width of the temporal graph's derivative. We exemplify its usefulness with meta theorems with respect to a temporal variant of first-order logic. The resulting logic expresses a wide range of temporal graph problems including a version of temporal cliques, an important notion when querying time series databases for community structures.

翻译：时态图是指顶点和边的存在性或属性随时间变化的图结构。当时间为离散时，时态图可定义为离散时间跨度（称为生存期）上的一系列静态图序列，也可定义为每条边与特定存活时间点集相关联的单一图。对于静态图，库尔塞勒定理断言：任何可在单子二阶逻辑中表达的图问题，均可在有界树宽图上以线性时间求解。本文首次提出适用于时态图单子二阶逻辑的库尔塞勒定理改进形式，该形式不显式依赖生存期作为参数。进而引入基于选定尺寸滑动时间窗口的导数概念，定义时态图导数的树宽与孪生宽度。通过针对时态一阶逻辑变体的元定理示例，论证了该框架的实用性。所得逻辑可表达广泛的时态图问题，包括时态团簇的变体——该概念在时序数据库中查询社区结构时具有重要意义。