Polygraphers generally interpret slow breathing to the result of a conscious effort to manipulate the "test" results. As such, it's considered to be a polygraph countermeasure (albeit one that is entirely ineffective and readily detectable).

The fact that you were specifically interrogated about drug use suggests that if indeed you failed, your failure is not necessarily attributable to slow breathing. Rather, it suggests that you may have reacted more strongly to the question(s) about illegal drug use than to the associated "control" question(s).