Definition
Formally, an edge cover of a graph G is a set of edges C such that each vertex in G is incident with at least one edge in C. The set C is said to cover the vertices of G. The following figure shows examples of edge coverings in two graphs.
A minimum edge covering is an edge covering of smallest possible size. The edge covering number is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings.
Note that the figure on the right is not only an edge cover but also a matching. In particular, it is a perfect matching: a matching M in which each vertex is incident with exactly one edge in M. A perfect matching is always a minimum edge covering.
Read more about this topic: Edge Cover
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