Many real-world networks, like transportation networks and social networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

翻译：许多现实世界的网络，如交通网络和社交网络，具有动态性，即边集可能随时间变化，但这些变化是已知的。这一行为由时态图模型刻画，该模型近期已成为理论计算机科学中的热门话题。该领域的核心开放问题是证明时态团中存在线性规模的时态支撑子图，即完全时态图的稀疏子图，确保通过时态路径实现所有节点对的可达性。目前，已知的最佳结果是存在边数为$\mathcal{O}(n\log n)$的时态支撑子图。我们在证明所有时态团中存在线性规模时态支撑子图方面取得了重要进展。我们改进了先前工作中使用的技术，并对其进行了大幅扩展和泛化，以提供更简单直观的$\mathcal{O}(n\log n)$边数界限证明。此外，我们利用新方法证明了一大类时态团（称为边枢轴图）具有线性规模的时态支撑子图。为进行对比，我们研究了不属于边枢轴图的其他时态团类别，引入两类此类图，并开发了在它们中也存在线性时态支撑子图的新技术。