  For overlay networks, the ability to recover from a variety of problems like
membership changes or faults is a key element to preserve their functionality.
In recent years, various self-stabilizing overlay networks have been proposed
that have the advantage of being able to recover from any illegal state.
However, the vast majority of these networks cannot give any guarantees on its
functionality while the recovery process is going on. We are especially
interested in searchability, i.e., the functionality that search messages for a
specific identifier are answered successfully if a node with that identifier
exists in the network. We investigate overlay networks that are not only
self-stabilizing but that also ensure that monotonic searchability is
maintained while the recovery process is going on, as long as there are no
corrupted messages in the system. More precisely, once a search message from
node $u$ to another node $v$ is successfully delivered, all future search
messages from $u$ to $v$ succeed as well. Monotonic searchability was recently
introduced in OPODIS 2015, in which the authors provide a solution for a simple
line topology.
  We present the first universal approach to maintain monotonic searchability
that is applicable to a wide range of topologies. As the base for our approach,
we introduce a set of primitives for manipulating overlay networks that allows
us to maintain searchability and show how existing protocols can be transformed
to use theses primitives.
  We complement this result with a generic search protocol that together with
the use of our primitives guarantees monotonic searchability.
  As an additional feature, searching existing nodes with the generic search
protocol is as fast as searching a node with any other fixed routing protocol
once the topology has stabilized.
