We discuss the connectivity of intersection spaces and the role of minimal energy points within these intersection spaces (minimal energy conical intersections or MECIs) in promoting nonadiabatic transitions. We focus on malonaldeyde as a specific example, where there is a low-lying three-state conical intersection. This three-state intersection is the global minimum on the bright excited electronic state, but it plays a limited role in population transfer in our ab initio multiple spawning (AIMS) simulations because the molecule must traverse a series of two-state conical intersections to reach the three-state intersection. Due to the differences in seam space dimensionality separating conventional (two-state) and three-state intersections, we suggest that dynamical effects arising directly from a three-state intersection may prove difficult to observe in general. We also use a newly developed method for intersection optimization with geometric constraints to demonstrate the connectivity of all the stationary points in the intersection spaces for malonaldehyde. This supports the conjecture that all intersection spaces are connected, and that three-state intersections play a key role in extending this connectivity to all pairs of states, e.g. the S1/S0 and S2/S1 intersection spaces.