Subnetwork mining is an essential issue in network analysis, with specific applications e.g. in biological networks, social networks, information networks and communication networks. Recent applications require the extraction of subnetworks (or patterns) involving several relations between the objects of interest, each such relation being given as a network. The complexity of a particular mining problem increases with the different nature of the networks, their number, their size, the topology of the requested pattern, the criteria to optimize. In this emerging field, our paper deals with two networks respectively represented as a directed acyclic graph and an undirected graph, on the same vertex set. The sought pattern is a longest path in the directed graph whose vertex set induces a connected subgraph in the undirected graph. This problem has immediate applications in biological networks, and predictable applications in social, information and communication networks. We study the complexity of the problem, thus identifying polynomial, NP-complete and APX-hard cases. In order to solve the difficult cases, we propose a heuristic and a branch-and-bound algorithm. We further perform experimental evaluation on both simulated and real data.