A temporal graph is a graph whose edges are available only at certain points in time. It is temporally connected if the nodes can reach each other by paths that traverse the edges chronologically (temporal paths). In general, temporal graphs do not always admit small subsets of edges that preserve connectivity (temporal spanners). In the case of temporal cliques, spanners of size $O(n\log n)$ are guaranteed. The original proof by Casteigts et al. [ICALP 2019] combines a number of techniques, one of which is dismountability. In a recent work, Angrick et al. [ESA 2024] simplified the proof and showed, among other things, that a one-sided version of dismountability can be used to replace the second part of the proof. In this paper, we revisit the dismountability principle. We characterizing the structure that a temporal clique has if it is not 1-hop dismountable, then not {1,2}-hop dismountable, and finally not {1,2,3}-hop dismountable. It turns out that if a clique is k-hop dismountable for any other k, then it must also be {1,2,3}-hop dismountable. Interestingly, excluding only 1-hop and 2-hop dismountability is already sufficient for reducing the spanner problem from cliques to bi-cliques. Put together with the strategy of Angrick et al., the entire $O(n \log n)$ result can now be recovered using only dismountability. An interesting by-product of our analysis is that any minimal counter-example to the existence of $4n$ spanners must satisfy the properties of non {1,2,3}-hop dismountable cliques. In the second part, we discuss connections between dismountability and pivotability. We show that recursively k-hop dismountable cliques are pivotable (and thus admits $2n$ spanners, whatever k). We define a family of labelings (called full-range) which force both dismountability and pivotability and that gives some evidence that large lifetimes could be exploited more generally.

翻译：时域图是一种边仅在特定时间点可用的图。若节点能通过按时间顺序遍历边的路径（时域路径）相互可达，则该图是时域连通的。一般而言，时域图并不总能存在保持连通性的小子边集（时域生成子图）。对于时域团的情形，可保证存在规模为 $O(n\log n)$ 的生成子图。Casteigts 等人 [ICALP 2019] 的原始证明结合了多种技术，其中一项是可拆卸性。在近期工作中，Angrick 等人 [ESA 2024] 简化了证明，并特别表明可使用单侧版本的可拆卸性替代原证明的第二部分。本文重新审视可拆卸性原理。我们刻画了时域团在非 1-跳可拆卸、非 {1,2}-跳可拆卸以及非 {1,2,3}-跳可拆卸时的结构特征。结果表明，若一个团对任意其他 k 值满足 k-跳可拆卸，则其必然也满足 {1,2,3}-跳可拆卸。有趣的是，仅排除 1-跳和 2-跳可拆卸性已足以将生成子图问题从团规约到二部团。结合 Angrick 等人的策略，现在仅通过可拆卸性即可完整复现 $O(n \log n)$ 的结果。我们分析的一个副产品是：任何关于 $4n$ 生成子图存在性的极小反例必然满足非 {1,2,3}-跳可拆卸团的性质。在第二部分，我们探讨可拆卸性与可枢轴性之间的关联。证明递归 k-跳可拆卸团具有可枢轴性（因而存在 $2n$ 生成子图，与 k 无关）。定义了一类强制同时满足可拆卸性与可枢轴性的标记方案（称为全范围标记），这为更广泛地利用长生命周期提供了依据。