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 a parameter proportional to the lifetime, or defined as the maximum number of time-edges incident with any vertex which in the worst case is higher than the lifetime. 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：时序图是指顶点和边的存在性或属性随时间变化的图。在离散时间情形下，时序图可定义为离散时间跨度（称为生命周期）上的静态图序列，或定义为每条边关联一组特定时刻（该边处于活跃状态）的单一图。对于静态图，库尔塞勒定理断言：在树宽有界的图上，任何可用一元二阶逻辑表达的图问题均可在线性时间内求解。我们提出了库尔塞勒定理在时序图上一元二阶逻辑中的首个适配方案，该方案无需显式依赖于与生命周期成正比的参数，也不依赖于定义为任意顶点关联的最大时间边数（该参数在最坏情况下大于生命周期）。随后，我们引入了在选定大小的滑动时间窗口上的导数概念，并定义了时序图导数的树宽与孪生宽度。我们通过关于时序一阶逻辑变体的元定理示例说明了其有效性。所得逻辑能表达广泛的时序图问题，包括时序团（查询时间序列数据库中社区结构时的重要概念）的某种形式。