Before revision:  Tracing techniques aim at reconstructing past history of contacts, but can face practical limits in an exponential growth of either testing or overly conservative quarantining. We show how this application of graph theory can be conveniently used to efficiently intercept new virus outbreaks, when they are still in their early stage.
After revision:  Contact tracing apps give the unprecedented possibility to reconstruct graphs of daily contacts, so the question is who should be tested? As human contact networks are known to exhibit community structure, in this paper we show that the Kemeny constant of a graph can be used to identify and analyze bridges between communities in a graph. Our "Kemeny indicator" is the change in Kemeny constant when a node or edge is removed from the graph. We show that testing individuals who are associated with large values of the Kemeny indicator can help in efficiently intercepting new virus outbreaks, when they are still in their early stage.