An edge cover is a subset of edges defined similarly to the vertex cover (Skiena 1990, p. 219), namely a collection of graph edges such that the union of edge endpoints corresponds to the entire vertex set of the graph. Therefore, only graphs with no isolated points have an edge cover.
An edge cover is a cover by edges inside a single graph, whereas a graph cover is a separate graph whose edges around each lifted vertex correspond one-to-one with the edges around its image in a base graph; equivalently, it maps locally bijectively onto the base graph.
A set of edges 
can be tested in the Wolfram Language
to see if it is an edge cover of a given graph using EdgeCoverQ[g,
e]. Precomputed edge covers for many named graphs can be looked up using GraphData[graph,
"EdgeCovers"].
An edge cover having the smallest possible number of edges for a given graph is known as a minimum edge cover. A minimum edge cover of a graph can be found in the Wolfram Language using FindEdgeCover[g]. An edge cover that does not contain any other edge cover as a proper subset is known as a minimal edge cover.
