In graph theory, an **edge cover** of a graph is a set of edges such that every vertex of the graph is incident to at least one edge in the set. In other words, an edge cover is a collection of edges that "covers" all vertices in the graph.
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An edge cover of a finite graph is a subset of its edges incident to every vertex. It exists if and only if there is no isolated vertex, with the empty graph covered by the empty set. A minimum-cardinality edge cover uses as few edges as possible; a minimum-weight edge cover minimizes their total weight. These objectives are related to maximum cardinality matching and maximum-weight matching, respectively, by the edge cover and matching identity and minimum-weight edge cover by matching savings.