Classical propositional logic is an appealing option for modelling argumentation but the computational viability of generating an argument is an issue. Here we propose ameliorating this problem by harnessing the notion of a connection graph to reduce the search space when seeking all the arguments for a claim from a knowledgebase. For a set of clauses, a connection graph is a graph where each node is a clause and each arc denotes that there exist complementary disjuncts in the pair of nodes. For a set of formulae in conjunctive normal form, we use the notion of the connection graph for the set of clauses obtained from the conjuncts in the formulae. When seeking arguments for a claim, we can focus our search on a particular subgraph of the connection graph that we call the focal graph. Locating this subgraph is relatively inexpensive in terms of computational cost. In addition, using (as the search space) the formulae of the initial knowledgebase, whose conjuncts relate to this subgraph, can substantially reduce the cost of looking for arguments. We provide a theoretical framework and algorithms for this proposal, together with some theoretical results and some preliminary experimental results to indicate the potential of the approach.