We study the problem of extracting a selective connector for a given set of query vertices $Q \subseteq V$ in a graph $G = (V,E)$. A selective connector is a subgraph of $G$ which exhibits some cohesiveness property, and contains the query vertices but does not necessarily connect them all. Relaxing the connectedness requirement allows the connector to detect multiple communities and to be tolerant to outliers. We achieve this by introducing the new measure of network inefficiency and by instantiating our search for a selective connector as the problem of finding the minimum inefficiency subgraph. We show that the minimum inefficiency subgraph problem is NP-hard, and devise efficient algorithms to approximate it. By means of several case studies in a variety of application domains (such as human brain, cancer, and food networks), we show that our minimum inefficiency subgraph produces high-quality solutions, exhibiting all the desired behaviors of a selective connector.