Temporal Cliques Admit Sparse Spanners

Authors Arnaud Casteigts , Joseph G. Peters , Jason Schoeters

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Author Details

Arnaud Casteigts
  • LaBRI, Université de Bordeaux, CNRS, Bordeaux INP, France
Joseph G. Peters
  • School of Computing Science, Simon Fraser University, Canada
Jason Schoeters
  • LaBRI, Université de Bordeaux, CNRS, Bordeaux INP, France


We thank Cyril Gavoille for advice on the presentation of these results.

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Arnaud Casteigts, Joseph G. Peters, and Jason Schoeters. Temporal Cliques Admit Sparse Spanners. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 134:1-134:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Let G=(V,E) be an undirected graph on n vertices and lambda:E -> 2^{N} a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph {G}=(G,lambda) is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity - a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of Theta(n^2)-dense temporal graphs which admit no temporal spanner of density o(n^2). The natural question is then whether sparse temporal spanners always exist in some classes of dense graphs. In this paper, we answer this question affirmatively, by showing that if the underlying graph G is a complete graph, then one can always find temporal spanners of density O(n log n). The best known result for complete graphs so far was that spanners of density binom{n}{2}- floor[n/4] = O(n^2) always exist. Our result is the first positive answer as to the existence of o(n^2) sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Dynamic graph algorithms
  • Dynamic networks
  • Temporal graphs
  • Temporal connectivity
  • Sparse spanners


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