How to find the insected area between the two neighbouring polygons using python programming

I am merging the neighbouring polygons based on the area calculation.Sometimes the polygon features are intersection at the edge or at one vertex only so when i am trying to merge those neighbouring polygon features it is closing the QGIS server. So i want to merge only the features that intersect atleast at the five vertices. So how can i find the intersected area between the two polygon features or how do i get the intersected area vertices?

I am using the following code to get the coordinates/vertices:

for f in layer.getFeatures():
  geom = f.geometry()
  print geom.asPolygon()

Is there any other way to solve this isssue?