Which of the following statements are TRUE? 1. The problem of determining whether there exists a cycle in an undirected graph is in P. 2. The problem of determining whether there exists a cycle in an undirected graph is in NP. 3. If a problem A is NP-Complete, there exists a non-deterministic polynomial time algorithm to solve A.

Let us analyze each statement:

1. Statement 1 is TRUE: Cycle detection in an undirected graph can be solved deterministically in $O(V + E)$ time using Depth First Search (DFS), Breadth First Search (BFS), or a Disjoint-Set (Union-Find) data structure. Since it runs in polynomial time on a deterministic machine, the problem is in $\text{P}$.

2. Statement 2 is TRUE: By definition, $\text{P} \subseteq \text{NP}$. Since cycle detection is in $\text{P}$, it is also in $\text{NP}$.

3. Statement 3 is TRUE: By definition, a language/problem $A$ is $\text{NP-Complete}$ if:

   • $A \in \text{NP}$, and

   • Every problem in $\text{NP}$ is polynomial-time reducible to $A$.

   Since $A \in \text{NP}$, there exists a non-deterministic Turing machine (or non-deterministic algorithm) that decides $A$ in polynomial time.

All three statements 1, 2, and 3 are TRUE. Therefore, (A) is the correct option.