Assume G is a weighted planar graph, nodes and edges in G are weighted, and K is a given constant. A is a decision problem with following description:    Does G contain a cycle with total  weight K ?What is the complexity of the counting version of A (#A)? (i.e the complexity of counting all cycles with total weight K) 
Complexity of counting weighted cycles in planar graphs?
cc.complexity theory;graph theory;counting complexity