Temporal cliques admit sparse spanners.

Let G = (V , E) be an undirected graph on n vertices and λ : E → 2 N a mapping that assigns to every edge a non-empty set of integer labels (discrete times when the edge is present). Such a labelled graph G = (G , λ) is 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 [17] asked whether, given such a temporally connected graph, a sparse subset of edges always exists whose labels suffice to preserve temporal connectivity – a temporal spanner. Axiotis and Fotakis [5] answered negatively by exhibiting a family of Θ (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 with O (n log ⁡ n) edges. [ABSTRACT FROM AUTHOR]