  The role of polymorphisms in determining the complexity of constraint
satisfaction problems is well established.
  In this context we study the stability of CSP complexity and polymorphism
properties under some basic graph theoretic constructions. As applications we
observe a collapse in the applicability of algorithms for CSPs over directed
graphs with both a total source and a total sink: the corresponding CSP is
solvable by the "few subpowers algorithm" if and only if it is solvable by a
local consistency check algorithm. Moreover, we find that the property of
"strict width" and solvability by few subpowers are unstable under first order
reductions. The analysis also yields a complete characterisation of the main
polymorphism properties for digraphs whose symmetric closure is a complete
graph.
