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Non-line-of-sight optical wireless transmission, operated in the unlicensed ultraviolet UV-C band, has been recently suggested as an alternative means of communication. However, due to limited coverage, relayed UV-C networks need to be deployed in order to supply communication services at large distances. In this paper, we consider a serial multi-hop UV-C network where the nodes are distributed at fixed positions on a given service interval. We adopt a suitable path loss model and derive analytical expressions for the node isolation probability assuming on-off keying and pulse position modulation formats. Moreover, we investigate the node density required to achieve connectivity for several geometrical transceiver configurations. The numerical results of this paper are of significant value for telecom researchers working toward a flexible UV-C network deployment in practice.