Path connectivity of line graphs and total graphs of complete bipartite graphs

•The minimum over all maximum number of edge-disjoint paths to connect any 3-subset θ of V(G) is given.•The compact upper bound of 3-path connectivity π3(G) for a general graph G is gotten.•The structural proprieties of line graphs L(Km,n) and total graphs T(Km,n) of complete bipartite graphs Km,n are deeply explored.•The 3-path connectivity π3(L(Km,n)) and π3(T(Km,n)) are determined.