Temporal Cliques Admit Sparse Spanners

TL;DR

This paper proves that in dense temporal graphs, specifically complete graphs, it is always possible to find sparse spanners with only O(n log n) edges that preserve temporal connectivity, answering a longstanding open question.

Contribution

It demonstrates the existence of sparse temporal spanners in dense graphs, specifically showing that complete graphs admit O(n log n) edge spanners that maintain temporal connectivity.

Findings

Complete graphs have O(n log n) sparse temporal spanners.

Sparse spanners preserve temporal connectivity in dense graphs.

First positive result for sparse spanners in dense temporal graphs.

Abstract

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer labels (times). Such a graph is {\em 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 \cite{KKK02} asked whether, given such a temporal graph, a {\em sparse} subset of edges always exists whose labels suffice to preserve temporal connectivity -- a {\em temporal spanner}. Axiotis and Fotakis \cite{AF16} answered negatively by exhibiting a family of $\Theta(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. In this paper, we give the first positive answer as to the existence of $o(n^2)$-sparse spanners in a dense class of temporal graphs, by showing (constructively) that if $G$ is a complete graph, then one can always find a temporal spanner of density $O(n \log n)$.