A proper subgraph of a connected linear graph is said to disconnect the graph if removing it leaves a disconnected graph. In this paper we characterize, in the following sense, the disconnecting subgraphs of a fixed connected graph. We define two distinct types of disconnecting subgraphs (isthmuses and articulators) which are minimal in the sense that no proper subgraph of either type can disconnect the graph. We then show that any disconnecting subgraph must contain either an isthmus or an articulator. We also define a set of subgraphs (called dense) which form a lattice. We show that the union of the minimal dense subgraphs contains all isthmuses and articulators. In terms of these subgraphs we investigate some of the consequences of assuming that a disconnecting subgraph must contain at least m points.