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This paper describes a node covering algorithm, i.e., a procedure for finding :I smallest set of nodes covering all edges of an arbitrary graph. The algorithm i.; based on the concept of a dual node-clique set, which allows us to identify partial covers associated with int.egw dual feasible solutions to the linear progr:utlminp equivalent of the node covering problem. An initial partid cover with the above property is first found by a labeling procedure. Another labeling procedure then successively modifies the dual node-clique set, so that more and more cdgrs : r c covered, i.e., the (primal) infeasibility of the solution i.; gradually rcdured, whilr, integrality and dual feasihilitp are preserved. When this cannot he continued, thc problem is partitioned and the procedure applied to the resulting aut)liroblrms. While the steps of the algorithm correspond to sequences of dual simples i)ivot.<, these are carried out implicitly, by labeling. The procedure is ilhist raited by c w mples, and some early computational experience is reported. We cnnclude \vit,h :I discussion of potential improvements and extensions.