  We address a specific case of the matroid intersection problem: given a set
of graphs sharing the same set of vertices, select a minimum cycle basis for
each graph to maximize the size of their intersection. We provide a
comprehensive complexity analysis of this problem, which finds applications in
chemoinformatics. We establish a complete partition of subcases based on
intrinsic parameters: the number of graphs, the maximum degree of the graphs,
and the size of the longest cycle in the minimum cycle bases. Additionally, we
present results concerning the approximability and parameterized complexity of
the problem.
