  Many man-made networks support each other to provide efficient services and
resources to the customers, despite that this support produces a strong
interdependency between the individual networks. Thus an initial failure of a
fraction $1-p$ of nodes in one network, exposes the system to cascade of
failures and, as a consequence, to a full collapse of the overall system.
Therefore it is important to develop efficient strategies to avoid the collapse
by increasing the robustness of the individual networks against failures. Here,
we provide an exact theoretical approach to study the evolution of the cascade
of failures on interdependent networks when a fraction $\alpha$ of the nodes
with higher connectivity in each individual network are autonomous. With this
pattern of interdependency we found, for pair of heterogeneous networks, two
critical percolation thresholds that depend on $\alpha$, separating three
regimes with very different network's final sizes that converge into a triple
point in the plane $p-\alpha$. Our findings suggest that the heterogeneity of
the networks represented by high degree nodes is the responsible of the rich
phase diagrams found in this and other investigations.
