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ZOJ Problem Set - 1790
The Geodetic Set Problem

Time Limit: 2 Seconds      Memory Limit: 65536 KB

Let G = (V, E) be a connected graph without loops and multiple edges, where V and E are the vertex and edge, respectively, sets of G. For any two vertices u, v in V, the distance between vertices u and v in G is the number of edges in a shortest u-v path. A shortest path between u and v is called a u-v geodesic. Let I(u, v) denote the set of vertices such that a vertex is in I(u, v) if and only if it is in some u-v geodesic of G and, for a set S <= V, I(S) = UNION(I(u, v)) (u, v in S). A vertex set D in graph G is called a geodetic set if I(D) = V. The geodetic set problem is to verify whether D is a geodetic set or not. We use Figure 3 as an example. In Figure 3, I(2, 5) = {2, 3, 4, 5} since there are two shortest paths between vertices 2 and 5. We can see that vertices 3 and 4 are lying on one of these two shortest paths respectively. However, I(2, 5) is not a geodetic set since I(2, 5) 6 != V. Vertex set {1, 2, 3, 4, 5} is intuitively a geodetic set of G. Vertex set D = {1, 2, 5} is also a geodetic set of G since vertex 3 (respectively, vertex 4) is in the shortest path between vertices 1 and 5 (respectively, vertices 2 and 5). Thus, I(D) = V. Besides, vertex sets {1, 3, 4} and {1, 4, 5} are also geodetic sets. However, D = {3, 4, 5} is not a geodetic set since vertex 1 is not in I(D).

Figure 3: A graph G.


The input consists of a given graph and several test cases. The first line contains an integer n indicating the number of vertices in the given graph, where 2 <= n <= 40. The vertices of a graph are labeled from 1 to n. Each vertex has a distinct label. The following n lines represent the adjacent vertices of vertex i, i = 1, 2, ..., n. For example, the second line of the sample input indicates that vertex 1 is adjacent with vertices 2 and 3. Note that any two integers in each line are separated by at least one space. After these n lines, there is a line which contains the number of test cases. Each test case is shown in one line and represents a given subset D of vertices. You have to determine whether D is a geodetic set or not.


For each test case, output 'yes' in one line if it is a geodetic set or 'no' otherwise.

Sample Input

2 3
1 3 4
1 2 5
2 5
3 4
1 2 3 4 5
1 2 5
2 4
1 3 4
1 4 5
3 4 5

Sample Output


Source: Asia 2003, Kaohsiung (Taiwan China)
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