Complexity of counting m-cycles in a graph with n nodes G is a planar graph with n nodes.
What are the complexity of following problems?

1-A: Does G contain an m-cycle? (m-cycle is a simple cycle with m nodes, m&lt; n)
2-B: complexity of counting all m-cycles in G, (Complexity of #A).

3- what is the complexity of A and B if G is an arbitrary given graph? 

Pointing to books and papers is also useful...
Perhaps this paper which deals with cycle lengths in planar graphs might be of value:

Li, Ming-Chu; Corneil, Derek G.; Mendelsohn, Eric (2000), "Pancyclicity and NP-completeness in planar graphs", Discrete Applied Mathematics 98 (3): 219–225.