Vertex covers of size greater than minimum

Question

If a graph G(v,e) has a minimum vertex cover of size k, does that mean that there are vertex covers of sizes between k and |v|?

For example, if G has 3 vertices, 2 edges, and has a VC of size 1, does that mean it also has VC's of sizes 2 and 3?

Answer 1

does that mean that there are vertex covers of sizes between k and |v|?

This is correct. Let $S \subset V$ be a minimum vertex cover. Then for any $A \subset (V \setminus S), S \cup A$ is also a vertex cover of $G$.

For example, if G = (3, 2)

What significance do $3$ and $2$ have here? Note that $V$ and $E$ are sets, not numbers. $V$ is the set of vertices, and $E$ is a set of $2$-element subsets of $V$.