1142 Maximal Clique (25 分)（图论，无向完全图）

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:

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

Sample Output:

Yes
Yes
Yes
Yes
Not Maximal
Not a Clique

生词

英文	解释
clique	小集团、派系

题目大意：

clique是一个点集，在一个无向图中，这个点集中任意两个不同的点之间都是相连的。maximal clique是一个clique，这个clique不可以再加入任何一个新的结点构成新的clique。点编号从1～nv，给出ne条边，以一对结点编号的方式给出。然后给出m条询问，每个询问是一个点集合，问这个点集合是否是maximal clique、是否是clique～

分析：

先判断是否是clique，即判断是否任意两边都相连；之后判断是否是maximal，即遍历所有不在集合中的剩余的点，看是否存在一个点满足和集合中所有的结点相连，最后如果都满足，那就输出Yes表示是Maximal clique～

原文链接：https://blog.csdn.net/liuchuo/article/details/79618797

题解

题目逻辑
1.判断是否为clique：若是，则转入2；否则直接输出Not a Clique结束。
2.判断是否为maximal：若是，则输出Yes；否则输出Not Maximal（是clique不是maximal）。
判断逻辑
1.判断clique：点集中所有不同的点是否全相连。
2.判断maximal：遍历不在点集中剩余的点，看 它 是否与 点集 中的点全不相连。

#include <bits/stdc++.h>

using namespace std;
int nv,ne;
int e[210][210];
int main()
{
#ifdef ONLINE_JUDGE
#else
    freopen("1.txt", "r", stdin);
#endif
    int a,b;
    cin>>nv>>ne;
    for(int i=0;i<ne;i++){
        cin>>a>>b;
        e[a][b]=e[b][a]=1;
    }
    int m,k;
    cin>>m;
    while(m--){
        cin>>k;
        vector<int> v(k);
        int hash[210]={0};
        int isclique=1,ismaximal=1;
        for(int i=0;i<k;i++){
            cin>>v[i];
            hash[v[i]]=1;
        }
        for(int i=0;i<k-1;i++){
            if(isclique==0) break;
            for(int j=i+1;j<k;j++){
                if(e[v[i]][v[j]]==0){
                    isclique=0;
                    cout<<"Not a Clique"<<endl;
                    break;
                }
            }
        }
        if(isclique==0) continue;
        for(int i=1;i<=nv;i++){
            if(hash[i]==0){
                for(int j=0;j<k;j++){
                    if(e[i][v[j]]==0) break;
                    if(j==k-1) ismaximal=0;
                }
            }
            if(ismaximal==0){
                cout<<"Not Maximal"<<endl;
                break;
            }
        }
        if(ismaximal==1) cout<<"Yes"<<endl;
    }
    return 0;
}