  We investigate a special case of hereditary property in graphs, referred to
as {\em robustness}. A property (or structure) is called robust in a graph $G$
if it is inherited by all the connected spanning subgraphs of $G$. We motivate
this definition using two different settings of dynamic networks. The first
corresponds to networks of low dynamicity, where some links may be permanently
removed so long as the network remains connected. The second corresponds to
highly-dynamic networks, where communication links appear and disappear
arbitrarily often, subject only to the requirement that the entities are
temporally connected in a recurrent fashion ({\it i.e.} they can always reach
each other through temporal paths). Each context induces a different
interpretation of the notion of robustness.
  We start by motivating the definition and discussing the two interpretations,
after what we consider the notion independently from its interpretation, taking
as our focus the robustness of {\em maximal independent sets} (MIS). A graph
may or may not admit a robust MIS. We characterize the set of graphs \forallMIS
in which {\em all} MISs are robust. Then, we turn our attention to the graphs
that {\em admit} a robust MIS (\existsMIS). This class has a more complex
structure; we give a partial characterization in terms of elementary graph
properties, then a complete characterization by means of a (polynomial time)
decision algorithm that accepts if and only if a robust MIS exists. This
algorithm can be adapted to construct such a solution if one exists.
