Final answer:
Connected vertices in a BFS tree differ by at most 1 layer can be proven using mathematical induction, counterexample, proof by contradiction, or direct proof.
Step-by-step explanation:
a. Mathematical induction:
To prove that connected vertices in a Breadth-first Search (BFS) tree differ by at most 1 layer, we can use mathematical induction. We first prove the base case by showing that the statement holds for the initial vertex. Then, we assume that the statement holds for a vertex at layer k and prove that it holds for a vertex at layer k+1. This ensures that the statement holds for all vertices in the BFS tree.
b. Counterexample:
A counterexample would be a graph where there is a vertex that is not connected to any other vertices. In this case, there would be no BFS tree, and the statement would not hold.
c. Proof by contradiction:
We can prove the statement by contradiction by assuming that there exists a BFS tree where connected vertices differ by more than 1 layer. We would then derive a contradiction by showing that this assumption leads to an inconsistency or impossibility.
d. Direct proof:
A direct proof involves verifying the statement directly using logical steps. In this case, we could consider a BFS tree and show that all connected vertices are at most 1 layer apart.