  A game-theoretic model for the study of dynamic networks is analyzed. The
model is motivated by communication networks that are subject to failure of
nodes and where the restoration needs resources. The corresponding two-player
game is played between "Destructor" (who can delete nodes) and "Constructor"
(who can restore or even create nodes under certain conditions). We also
include the feature of information flow by allowing Constructor to change
labels of adjacent nodes. As objective for Constructor the network property to
be connected is considered, either as a safety condition or as a reachability
condition (in the latter case starting from a non-connected network). We show
under which conditions the solvability of the corresponding games for
Constructor is decidable, and in this case obtain upper and lower complexity
bounds, as well as algorithms derived from winning strategies. Due to the
asymmetry between the players, safety and reachability objectives are not dual
to each other and are treated separately.
