Let G(V,E) be a simple, undirected graph. A vertex cover of G is a subset V′ ⊆ V such that for every (u,v) ∈ E, u ∈ V′ or v ∈ V′. Let the size of the smallest vertex cover in G be k. Let S be any vertex cover of size k. For a vertex v ∈ V, which of the following constraints will always ensure that v ∈ S?

The correct option is: The degree of v is at least k + 1.

Let us understand why this constraint always ensures that the vertex $v$ must be in any vertex cover $S$ of the minimum size $k$.

By definition, a vertex cover $S$ of a graph $G=(V,E)$ is a subset of vertices such that every edge in the graph has at least one of its endpoints in $S$.
Here, we are given that the size of the smallest vertex cover is $k$, and $S$ is any vertex cover of size $k$, so $\left|S\right|=k$.

Let the degree of a vertex $v$ be denoted by $\mathrm{deg}\left(v\right)$. The degree of $v$ is the number of edges incident to $v$.
Suppose that $\mathrm{deg}\left(v\right)\ge k+1$. This means there are at least $k+1$ distinct edges incident to $v$, say $(v,u_1),(v,u_2),...,(v,u_d)$ where $d\ge k+1$ and $u_1,u_2,...,u_d$ are distinct neighbors of $v$.

We want to prove that $v$ must belong to $S$. Let us prove this by contradiction.
Assume for contradiction that $v\notin S$.

Since $S$ is a valid vertex cover, it must cover all the edges incident to $v$. That is, for every edge $(v,u_i)$ for $i=1,2,...,d$, at least one endpoint of the edge must be in $S$.
Because we assumed $v\notin S$, the other endpoint $u_i$ must be in $S$ for every $i=1,2,...,d$ to cover each of these incident edges.

Therefore, we must have:
$\{u_1,u_2,...,u_d\}\subseteq S$
Since all $u_i$ are distinct, the size of the vertex cover $S$ must satisfy:
$\left|S\right|\ge d$
Substituting $d\ge k+1$, we get:
$\left|S\right|\ge k+1$
But this contradicts our initial fact that the size of $S$ is $k$ ($\left|S\right|=k$).

Since the assumption $v\notin S$ leads to a contradiction, the vertex $v$ must be in $S$.
Thus, if the degree of $v$ is at least $k+1$, $v$ is guaranteed to be in any vertex cover of size $k$.