Dismountability in Temporal Cliques Revisited

2025-02-03

AI Summary

This paper investigates the construction of temporal spanners in temporal complete graphs, focusing on the critical role of *dismountability*—a structural property preserving temporal clique connectivity. Methodologically, it employs structural induction and reduction analysis to fully characterize the combinatorial structure of 1-, 2-, and 3-hop *undismountable* temporal cliques—the first such complete characterization. It proves that excluding only 1- and 2-hop dismountability suffices to reduce spanner construction to a bipartite graph problem. Furthermore, it establishes an equivalence between dismountability and *pivotability*, showing that any *k*-hop recursively dismountable clique is necessarily pivotable. Leveraging this equivalence, the paper reformulates the existence proof for *O(n log n)*-size temporal spanners purely in terms of dismountability. Finally, it identifies that minimal counterexamples to the conjectured 2*n*-edge upper bound must be {1,2,3}-hop undismountable cliques—providing crucial structural evidence supporting the 2*n*-size spanner conjecture.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 $O(nlog n)$ 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.

Problem

• Temporal Graphs
• Decomposability
• Pivotability

Innovation

• k-jump removable
• pivotability
• labeling technique