检测一个由链表表示的图中3个相邻顶点的循环

Question

给定一个无向图，其边数多于顶点数，并用相邻顶点的链表表示。如何检测是否存在3个相邻顶点的循环，时间复杂度是多少？

示例图：

1->2->5
2->3->1->4
3->2->4
4->2->5->3
5->1->4

存在3个相邻顶点的循环2->3->4->2

Answer 1

这个问题可以解决，使用算法深度优先搜索:

#include <iostream>
#include <vector>
#include <set>
#include <algorithm>
using namespace std;
const int maximumSize=40;
vector<vector<int>> visited(maximumSize, vector<int>(maximumSize, 0));
vector<int> graph[maximumSize], closedContour, temporary;
int vertices, edges;
set<vector<int>> contours;
void showContentSetVector(set<vector<int>> input)
{
    for(auto iterator=input.begin(); iterator!=input.end(); ++iterator)
    {
        for(auto item : *iterator)
        {
            cout<<item<<", ";
        }
        cout<<endl;
    }
    return;
}
bool compare(int i,int j)
{
    return (i<j);
}
void createGraph()
{
    cin>>vertices>>edges;
    int vertex0, vertex1;
    for(int i=1; i<=edges; ++i)
    {
        cin>>vertex0>>vertex1;
        graph[vertex0].push_back(vertex1);
        graph[vertex1].push_back(vertex0);
    }
    return;
}
void depthFirstSearch(int initial, int current, int previous)
{
    if(visited[initial][current]==1)
    {
        for(int i=0; i<temporary.size(); ++i)
        {
            if(temporary[i]==current)
            {
                for(int j=i; j<temporary.size(); ++j)
                {
                    closedContour.push_back(temporary[j]);
                }
            }
        }
        sort(closedContour.begin(), closedContour.end(), compare);
        contours.insert(closedContour);
        closedContour.clear();
        return;
    }
    visited[initial][current]=1;
    temporary.push_back(current);
    for(int next : graph[current])
    {
        if(next==previous)
        {
            continue;
        }
        depthFirstSearch(initial, next, current);
    }
    temporary.pop_back();
    return;
}
void solve()
{
    createGraph();
    for(int vertex=1; vertex<=vertices; ++vertex)
    {
        temporary.clear();
        depthFirstSearch(vertex, vertex, -1);
    }
    cout<<"contours <- ";
    showContentSetVector(contours);
    return;
}
int main()
{
    solve();
    return 0;
}

结果如下：

contours <- 
1: 1, 2, 3, 4, 5, 
2: 1, 2, 4, 5, 
3: 2, 3, 4,

正如您在提供的循环（闭合轮廓）中看到的那样，您在问题中提到了该轮廓，由 3 个相邻顶点组成：vertices={2, 3, 4} .

因为您提到您的 图是无向的 我认为您的示例图必须如下所示：

1:[2, 5],
2:[1, 3, 4],
3:[2, 4],
4:[2, 3, 5],
5:[1, 4].

检测一个由链表表示的图中3个相邻顶点的循环

Detect a cycle of 3 adjacent vertices in a graph represented by a linked list

data-structures

linked-list

time-complexity

graph-theory