In the case of k-approval control by adding voters for k > **4, we modify the reduction as follows. Each**
voter in V now approves of k candidates in S**, and thus without loss of generality, we now assume that** k divides 3m**. In each voter** viin V
′, we will approve of k − 4 buffer candidates, {b1, . . . , bk−4} **in addition to**
{p, ti,1, ti,2, ti,3}**. Following an addition of voters, each buffer candidate will have exactly the same number**
of approvals as p**, and will this not affect the status of** p.

Theorem 17 k**-veto elections are computationally resistant to constructive control by adding voters for** k ≥ 3.

Proof: We begin by showing that 3-veto control by adding voters is NP-hard and extend it to k**-veto for** k > **3. Let**
S = {s1**, . . . , s**3m}
and T1 = {t1,1, t1,2, t1,3}, . . . , Tn = {tn,1, tn,2, tn,3}
be an instance of X3C. We define our candidate set to be C = {p, p′, p′′} ∪ {s1**, . . . , s**3m}.

Our initial voter set V will consist of one vote, which vetoes {p, p′, p′′}.

For each set Ti = {ti,1, ti,2, ti,3}, we add a voter vi **to our unestablished voter set** V
′ **who vetoes**
{ti,1, ti,2, ti,3}. We show that a manipulation by adding at most m **voters exists iff there is an exact covering**
for this instance of X3C.

Consider an exact 3-set covering. Adding the m **votes corresponding to the covering gives each candidate**
exactly one veto. p is thus made a winner. In contrast, p **can only be made a winner by adding one veto to**
each candidate in si. Since only m **votes are added, this corresponds to a solution of the X3C instance.**
Theorem 18 k**-approval elections are computationally resistant to constructive control by deleting voters** for k ≥ 3.

Proof: The reduction for this case is similar to that of bribery. In this case, **instead of bribing the voters**
to vote for {p, p′, p′′} **we simply delete them.**
Theorem 19 k**-veto elections are computationally resistant to constructive control by deleting voters for** k ≥ 4.

Proof: We note that the construction in the proof that k**-veto is computationally resistant to bribery**
in the unweighted and unpriced case also applies to constructive control by deleting voters. In this case, instead of bribing the voters to vote for buffer candidates, we simply delete them.