# [Math] Relation between sizes of matching, edge cover, independent set and vertex cover

graph theory

1. From Wikipedia:

A perfect matching is also a minimum-size edge cover. Thus, the size
of a maximum matching is no larger than the size of a minimum edge
cover.

I know that a perfect matching is a maximum and hence maximal
matching. But I don't understand why "the size of a maximum matching is no
larger than the size of a minimum edge cover"?

2. Is the size of any matching no
larger than the size of any edge cover? If Part 1 is true, then I think part one implies this part?
3. Similar questions for independent sets and vertex covers.

I understand that the complement of an independent set is a vertex
cover, and the complement of a vertex cover is an independent set.

• Is there a definite answer to which is no greater than which, the
size of a maximum independent set and the size of a minimum vertex
cover?

• Similar question for the size of any independent set and the size of
any vertex cover?

Thanks!

For vertex covers and independent sets, the complete bipartite graph $K_{1,3}$ has a vertex cover of size 1 and an independent set of size 3. The cycle $C_5$ has an independent set of size 2, but any vertex cover has size at least three. So there is no simple relation between the minimum size of a vertex cover and the maximum size of an independent set.