Let $G = (V, E)$ be a simple undirected graph, and $s$ be a particular vertex in it called the source. For $x \in V$, let $d(x)$ denote the shortest distance in $G$ from $s$ to $x$. A breadth first search (BFS) is performed starting at $s$. Let $T$ be the resultant BFS tree. If $(u,v)$ is an edge of $G$ that is not in $T$, then which one of the following CANNOT be the value of $d(u) - d(v)$?

In an unweighted, undirected graph $G = (V, E)$, the shortest path distance from a source vertex $s$ satisfies the triangle inequality for every edge $(u, v) \in E$:

This means that for any edge $(u, v)$ in $G$ (whether it belongs to the BFS tree $T$ or is a cross/non-tree edge), the distance labels of $u$ and $v$ can differ by at most $1$. Thus, the possible values of $d(u) - d(v)$ are strictly:

Therefore, $d(u) - d(v)$CANNOT be $2$.

Hence, the correct option is (D).