E - Magical Ornament

Score : $500$ points

Problem Statement

There are $N$ kinds of magical gems, numbered $1, 2, \ldots, N$, distributed in the AtCoder Kingdom.
Takahashi is trying to make an ornament by arranging gems in a row.
For some pairs of gems, we can put the two gems next to each other; for other pairs, we cannot. We have $M$ pairs for which the two gems can be adjacent: (Gem $A_1$, Gem $B_1$), (Gem $A_2$, Gem $B_2$), $\ldots$, (Gem $A_M$, Gem $B_M$). For the other pairs, the two gems cannot be adjacent. (Order does not matter in these pairs.)
Determine whether it is possible to form a sequence of gems that has one or more gems of each of the kinds $C_1, C_2, \dots, C_K$. If the answer is yes, find the minimum number of stones needed to form such a sequence.

Constraints

• All values in input are integers.
• $1 ≤ N ≤ 10^5$
• $0 ≤ M ≤ 10^5$
• $1 ≤ A_i < B_i ≤ N$
• If $i ≠ j$, $(A_i, B_i) ≠ (A_j, B_j)$.
• $1 ≤ K ≤ 17$
• $1 ≤ C_1 < C_2 < \dots < C_K ≤ N$

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$

$A_2$ $B_2$

$\hspace{7mm}\vdots$

$A_M$ $B_M$

$K$

$C_1$ $C_2$ $\cdots$ $C_K$

Output

Print the minimum number of stones needed to form a sequence of gems that has one or more gems of each of the kinds $C_1, C_2, \dots, C_K$.
If such a sequence cannot be formed, print -1 instead.

4 3
1 4
2 4
3 4
3
1 2 3

5

For example, by arranging the gems in the order $[1, 4, 2, 4, 3]$, we can form a sequence of length $5$ with Gems $1, 2, 3$.

4 3
1 4
2 4
1 2
3
1 2 3

-1

10 10
3 9
3 8
8 10
2 10
5 8
6 8
5 7
6 7
1 6
2 4
4
1 2 7 9

11

For example, by arranging the gems in the order $[1, 6, 7, 5, 8, 3, 9, 3, 8, 10, 2]$, we can form a sequence of length $11$ with Gems $1, 2, 7, 9$.