Let $L \subseteq \{0, 1\}^*$ be an arbitrary regular language accepted by a minimal DFA with $k$ states. Which one of the following languages must necessarily be accepted by a minimal DFA with $k$ states?

Let $M = (Q, \Sigma, \delta, q_0, F)$ be the minimal DFA accepting $L$ with $|Q| = k$ states.

The complement of $L$ is $\overline{L} = \{0, 1\}^* - L$. The DFA accepting $\overline{L}$ is constructed by simply inverting the set of final states:

1. Reachability: Since all states in $M$ are reachable from $q_0$ and the transition function $\delta$ is unchanged, all states in $M^{\prime}$ remain reachable.

2. Distinguishability: Any two distinct states $p, q \in Q$ that are distinguished by a string $w \in \Sigma^*$ in $M$ (meaning one leads to an accepting state and the other to a non-accepting state) will still be distinguished by $w$ in $M^{\prime}$ (since the acceptance status of all states is inverted).

Therefore, $M^{\prime}$ is also minimal and has precisely $k$ states.

For the other operations (difference with a singleton set, union with a singleton set, concatenation), the number of states in the minimal DFA can change depending on $L$.

Hence, the correct option is (C).