<p>In this thesis, for symplectic and split odd special orthogonal groups over a p-adic field, we provide an extensive account of the intersection of local Arthur packets. More specifically, following Atobe's reformulation of Moeglin's construction of local Arthur packets, we give precise algorithms to determine whether a given representation lies in any local Arthur packet. Furthermore, if the representation does lie in a local Arthur packet, we give a systematic approach to determine all the other local Arthur packets which also contain the representation. These algorithms are based on certain operators which act on certain construction data coming from Atobe's parametrization of the local Arthur packets.</p>