[PAT] 1142 Maximal Clique（25 分）

1142 Maximal Clique（25 分）

A clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. A maximal clique is a clique that cannot be extended by including one more adjacent vertex. (Quoted from https://en.wikipedia.org/wiki/Clique_(graph_theory)

Now it is your job to judge if a given subset of vertices can form a maximal clique.

Input Specification:

Each input file contains one test case. For each case, the first line gives two positive integers Nv (≤ 200), the number of vertices in the graph, and Ne, the number of undirected edges. Then Ne lines follow, each gives a pair of vertices of an edge. The vertices are numbered from 1 to Nv.

After the graph, there is another positive integer M (≤ 100). Then M lines of query follow, each first gives a positive number K (≤ Nv), then followed by a sequence of K distinct vertices. All the numbers in a line are separated by a space.

Output Specification:

For each of the M queries, print in a line Yes if the given subset of vertices can form a maximal clique; or if it is a clique but not a maximal clique, print Not Maximal; or if it is not a clique at all, print Not a Clique.

Sample Input:

Sample Output:

Yes
Yes
Yes
Yes
Not Maximal
Not a Clique

题意：
clique是一个无向图的点的子集，clique里每两个不同的点都是相连的。最大clique是不能扩展更多相连点的clique。
判断给定的点的子集是否为一个最大的clique。

思路：
将输入存在一个二维数组（矩阵）内，对于每一个查询，先判断是否为clique（每两个点对应的二维数组为1），再判断是否为max clique（遍历给定点之外的点是否和给定的点都有连线）。

题解：

 1 #include<cstdlib>
 2 #include<cstdio>
 3 #include<vector>
 4 using namespace std;
 5 int vertex[210][210];
 6 
 7 int main() {
 8     int nv, ne, m;
 9     scanf("%d %d", &nv, &ne);
10     for (int i = 0; i < ne; i++) {
11         int a, b;
12         scanf("%d %d", &a, &b);
13         vertex[a][b] = vertex[b][a] = 1;
14     }
15     scanf("%d", &m);
16     for (int i = 0; i < m; i++) {
17         int k;
18         scanf("%d", &k);
19         vector<bool> v(nv+1);
20         vector<int> verts;
21         for (int j = 0; j < k; j++) {
22             int t;
23             scanf("%d", &t);
24             verts.push_back(t);
25             v[t] = true;
26         }
27         //判断是否为clique
28         //只要给定查询中的每两个点对应矩阵上的值为0，说明它不是clique
29         bool isClique = true;
30         for (int j = 0; j < verts.size(); j++) {
31             for (int r = j + 1; r < verts.size(); r++) {
32                 if (vertex[verts[j]][verts[r]] == 0) {
33                     printf("Not a Clique\n");
34                     isClique = false;
35                     break;
36                 }
37             }
38             if (!isClique) break;
39         }
40 
41         if (!isClique) continue;
42         
43         //如果是clique，
44         //并且还存在给定之外的点与给定的所有点对应矩阵的值为1
45         //说明其不是max clique
46         bool notMaximal = false;
47         for (int j = 1; j <= nv; j++) {
48             if (v[j] == true) continue;
49             bool isMaxClique = false;
50             for (int r = 0; r < verts.size(); r++) {
51                 if (vertex[j][verts[r]] == 0) {
52                     isMaxClique = true;
53                     break;
54                 }
55             }
56             if (!isMaxClique) {
57                 printf("Not Maximal\n");
58                 notMaximal = true;
59                 break;
60             }
61         }
62         if (!notMaximal) printf("Yes\n");
63     }
64     return 0;
65 }