Consider the Deterministic Finite-state Automaton (DFA) $\mathcal{A}$ shown below. The DFA runs on the alphabet $\{0, 1\}$, and has the set of states $\{s, p, q, r\}$, with $s$ being the start state and $p$ being the only final state. Which one of the following regular expressions correctly describes the language accepted by $\mathcal{A}$?

To find the regular expression describing the language accepted by the DFA:

1. Initial Transition:

   • From the start state $s$, reading $1$ transitions to the accepting state $p$.

   • Reading $0$ from $s$ transitions to state $r$, which is a trap/dead state (it loops on both $0$ and $1$ and has no outgoing transitions to any accepting state).

   • Thus, every accepted string must start with $1$.

2. Transitions from the Accepting State $p$:

   • On reading $0$, the automaton stays in $p$ (a self-loop on $0$).

   • On reading $1$, the automaton moves to state $q$.

3. Transitions from State $q$:

   • From state $q$, reading $1$ transitions back to $p$. This forms a path $p \xrightarrow{1} q \xrightarrow{1} p$, corresponding to the string $11$.

   • From state $q$, reading $0$ transitions to the trap state $r$.

4. Accepted Language:

   • Once at state $p$, any combination of the self-loop $0$ or the cycle $11$ keeps the automaton in state $p$ or returns it to $p$.

   • Hence, after the initial prefix $1$, any sequence formed by $0$ or $11$ is accepted, which corresponds to the regular expression:

     $$1(0 + 11)^*$$

Therefore, (C) is the correct option.