Temporal graphs provide a useful model for many real-world networks. Unfortunately the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which describe tractable cases by simultaneously restricting the underlying structure and the times at which edges appear in the graph. These all rely on the temporal graph being sparse in some sense. We introduce temporal analogues of three increasingly restrictive static graph parameters -- cliquewidth, modular-width and neighbourhood diversity -- which take small values for highly structured temporal graphs, even if a large number of edges are active at each timestep. The computational problems solvable efficiently when the temporal cliquewidth of the input graph is bounded form a subset of those solvable efficiently when the temporal modular-width is bounded, which is in turn a subset of problems efficiently solvable when the temporal neighbourhood diversity is bounded. By considering specific temporal graph problems, we demonstrate that (up to standard complexity theoretic assumptions) these inclusions are strict.

翻译：时序图为许多现实网络提供了有用模型。然而，我们可能在此类图上考虑的大多数算法问题都难以处理。近期在定义结构参数方面取得了进展，这些参数通过同时限制基础结构和边在图中的出现时间来刻画可处理情形。这些参数均依赖于时序图在某种意义上是稀疏的这一前提。我们引入了三种严格性递增的静态图参数——团宽度、模宽度和邻域多样性——的时序对等物，对于高度结构化的时序图，即使每个时间步有大量边处于活动状态，这些参数仍取较小值。当输入图的时序团宽度有界时，可高效求解的计算问题构成当输入图的时序模宽度有界时可高效求解问题的子集，而后者又进一步构成当输入图的时序邻域多样性有界时可高效求解问题的子集。通过考虑具体的时序图问题，我们证明了（在标准复杂性理论假设下）这些包含关系是严格的。