Biological networks are commonly used to model molecular activity within the cell. Recent experimental studies have shown that the detection of conserved subnetworks across several networks, coming from di erent organisms, may allow the discovery of disease pathways and prediction of protein functions. There already exist automatic methods that allow to search for conserved subnetworks using networks alignment; unfortunately, these methods are limited to networks of same type, thus having the same graph representation. Towards overcoming this limitation, a uni ed framework for pairwise comparison and analysis of networks with di erent graph representations (in particular, a directed acyclic graph D and an undirected graph G over the same set of vertices) is presented in [4]. We consider here a related problem called k-DAGCC: given a directed graph D and an undirected graph G on the same set V of vertices, and an integer k, does there exist sets of vertices V1; V2; : : : Vk0 , k0 k such that, for each 1 i k0, (i) D[Vi] is a DAG and (ii) G[Vi] is connected ? Two variants of k-DAGCC are of interest: (a) the Vis must form a partition of V , or (b) the Vis must form a cover of V . We study the computational complexity of both variants of k-DAGCC and, depending on the constraints imposed on the input, provide several polynomial-time algorithms, hardness and inapproximability results.