Let $G$ be a simple undirected graph with two vertices $a$, $b$ and an edge ($a$, $b$). Clearly $T_D$ of this graph would be ($a$, $b$). From the given definition of cross edges, ($a$, $b$) is also a cross edge because $a$ or $b$, neither is an ancestor of the other in $T_D$. Therefore statement (I) must not necessarily be true.
Consider the following graph $G(V,E)$.
V={$a, b, c$}, E= {($a,b$), ($b,c$), ($a,c$)}
One of the breadth first search tree, $T_B$ = {($a,b$), ($a,c$)}
For edge ($b,c$) of $G$, $b$ and c vertices both are at the same depth in $T_B$. Depth difference for vertices $b$ and $c$ in $T_B$ equals to 0. Therefore statement (II) must not necessarily be true.
Hence correct answer is (D).