Static vacuum spacetimes with one compact imension include black holes with localized horizons but also uniform and non-uniform black
strings where the horizon wraps over the compact dimension. The set of these black objects with other bubble type solutions is an example for the breakdown of black hole uniqueness in higher
dimensions, and completing the phase diagram will give qualitative understanding of the topology change between the black objects. I discuss how recent numerical works have revealed many important aspects related to this issue,
and I present some new numerical solutions for these localized black holes.
Combined with previous non-uniform string results, the solutions provide good evidence that the black hole and non-uniform string branches join at a topology changing solution.
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