Dismountability in Temporal Cliques Revisited

TL;DR

The paper advances a systematic, structural treatment of dismountability in temporal cliques, proving that if a clique is $k$-hop dismountable for any $k$, then it is already $\{1,2,3\}$-hop dismountable, and that excluding $1$- and $2$-hop dismountability suffices to reduce the sparse spanner problem to bi-cliques. Building on this, a unified, constructive approach recovers the known $O(n\log n)$-sparse spanners for temporal cliques using only dismountability, and it connects dismountability to pivotability, including results on full-range graphs and their potential to yield sparser spanners. The work also characterizes the precise structural properties of non-dismountable cliques and shows that lifetime considerations (full-range graphs) reinforce pivotability and higher sparsity. Overall, the paper consolidates dismountability as a central technique for sparse temporal spanners and highlights open questions about linear-size spanners and possible relaxations of dismountability.

Abstract

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.

Figures (11)

• Figure 1: A temporally connected graph $\mathcal{G}\xspace$ and two temporal spanners $\mathcal{G}\xspace'$ and $\mathcal{G}\xspace"$.
• Figure 2: A recursively $1$-hop dismountable graph and the resulting spanner (Fig. from CPS19).
• Figure 3: Illustration of the principle of dismountability and $k$-hop dismountability, where $-$ and $+$ denote the local earliest and latest edges, respectively.
• Figure 4: Node $v$ is 2-hop dismountable.
• Figure 6: Reciprocity in a non $\{1,2\}$-hop dismountable graph.

Theorems & Definitions (20)

• Theorem 1: $1$-hop dismountability CPS19
• Lemma 2
• Lemma 3
• Corollary 4
• Lemma 5
• Lemma 6: Reciprocity
• Lemma 7
• Theorem 8
• Corollary 9
• Theorem 10: Summary of non $\{1,2\}$-hop dismountability