  We present a cascading failure model of two interdependent networks in which
functional nodes belong to components of size greater than or equal to $s$. We
find theoretically and via simulation that in complex networks with random
dependency links the transition is first-order for $s\geq 3$ and second-order
for $s=2$. We find for two square lattices with a distance constraint $r$ in
the dependency links that increasing $r$ moves the system from a regime without
a phase transition to one with a second-order transition. As $r$ continues to
increase the system collapses in a first-order transition. Each regime is
associated with a different structure of domain formation of functional nodes.
