# Temporal Cliques admit Sparse Spanners

Let G=(G,λ) be a labeled graph on n vertices with λ:E_G→N a locally injective mapping that assigns to every edge a unique integer label. The label is seen as a discrete time when the edge is present. This graph is temporally connected if a path exists with increasing labels from every vertex to every other vertex. We investigate a question by Kempe, Kleinberg, and Kumar (JCSS 2002), asking whether sparse subsets of edges can always be found that preserve temporal connectivity if the rest is removed. We call such subsets temporal spanners of G. Despite the simplicity of the model, no significative progress has been made so far on this question; the only known result is that Ω(n) edges can always be removed if G is a complete graph. In this paper, we settle the question (affirmatively) for complete graphs, showing that all edges but a sub-quadratic amount of them (namely O(n n)) can always be removed, making the first significant progress towards the original question. This is done in several gradual steps. First, we observe that the existing argument for removing Θ(n) edges can be generalized to removing Θ(n^2) edges (namely, a sixth of the edges). Then, building on top of a gradual set of techniques, we establish that a quarter of the edges can always be removed, then half of the edges, and eventually all but O(n n) edges. These results are robust in the sense that they extend, with reasonable assumptions, to general graph models where the labels may not be locally unique and every edge may have several labels. We conjecture that sparse spanners always exist in the general case (G not being complete) and we hope that some of our techniques might be used in this perspective.

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