Hello! Today we will learn about bipartite graphs<\/strong>. A bipartite graph is a graph in which the set of vertices can be divided into two subsets, and there are no edges between vertices within the same subset. This topic is one of the frequently posed questions in coding tests. In the next session, we will explore the algorithms and coding needed to determine if a graph is bipartite.<\/p>\n The problem is to determine whether the given undirected graph is bipartite. Starting from a vertex, we need to check whether we can visit all vertices while satisfying the bipartite graph conditions.<\/p>\n Given the following graph, determine whether it is a bipartite graph.<\/p>\n Looking at the graph above, we can see if it can be divided into two sets. Set A is {1, 4, 5, 6} and Set B is {2, 3}. This graph is indeed a bipartite graph.<\/p>\n One method we can use to confirm if a graph is bipartite is Breadth-First Search (BFS)<\/strong> or Depth-First Search (DFS)<\/strong>. Both methods can be employed to traverse the graph while coloring each vertex (or grouping them) to verify the bipartite graph conditions.<\/p>\n Let’s take a look at the C++ code to determine if this undirected graph is bipartite.<\/p>\n This code takes the vertices and edges of the given graph as input and determines whether the graph is bipartite. The isBipartite<\/strong> function checks if all adjacent vertices are of different colors.<\/p>\n When running the code, an example of inputting edges is as follows.<\/p>\n When given the above input, the output will be as follows.<\/p>\n In conclusion, we learned how to determine if a graph is bipartite using C++. Understanding the basic principle of solving problems using BFS or DFS can be applied to various graph problems. I hope this helps your preparation for coding tests!<\/p>\nProblem Description<\/h2>\n
Problem Example<\/h3>\n
\nVertices: 6\nEdges: \n1 - 2\n1 - 3\n2 - 4\n2 - 5\n3 - 6\n<\/pre>\n
Problem Solving Strategy<\/h2>\n
Algorithm Steps<\/h3>\n
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C++ Code Implementation<\/h2>\n
\n#include <iostream>\n#include <vector>\n#include <queue>\n\nusing namespace std;\n\nconst int MAX = 1000; \/\/ Maximum number of vertices\nvector<int> graph[MAX];\nint color[MAX]; \/\/ Color setting: -1 is unvisited, 0 and 1 are groups\n\nbool isBipartite(int start) {\n queue<int> q;\n q.push(start);\n color[start] = 0; \/\/ Color the starting vertex\n\n while (!q.empty()) {\n int v = q.front();\n q.pop();\n\n for (int neighbor : graph[v]) {\n \/\/ If the neighboring vertex has not been visited, color it.\n if (color[neighbor] == -1) {\n color[neighbor] = 1 - color[v]; \/\/ Assign a different color than the current vertex\n q.push(neighbor);\n }\n \/\/ If the neighboring vertex has the same color as the current vertex, it is not a bipartite graph.\n else if (color[neighbor] == color[v]) {\n return false; \/\/ Violates the bipartite graph conditions\n }\n }\n }\n return true;\n}\n\nint main() {\n int v, e;\n cout << \"Enter the number of vertices and edges: \";\n cin >> v >> e;\n\n \/\/ Initialize the graph\n for (int i = 0; i < MAX; i++) {\n color[i] = -1;\n }\n\n cout << \"Input edges (format: a b): \" << endl;\n for (int i = 0; i < e; i++) {\n int a, b;\n cin >> a >> b;\n graph[a].push_back(b);\n graph[b].push_back(a); \/\/ Undirected graph\n }\n\n bool result = true;\n for (int i = 1; i <= v; i++) {\n if (color[i] == -1) {\n \/\/ If this vertex is unvisited, check if it is bipartite\n if (!isBipartite(i)) {\n result = false;\n break;\n }\n }\n }\n\n if (result) {\n cout << \"This graph is a bipartite graph.\" << endl;\n } else {\n cout << \"This graph is not a bipartite graph.\" << endl;\n }\n\n return 0;\n}\n<\/code><\/pre>\nCode Explanation<\/h2>\n
Variable and Structure Explanation<\/h3>\n
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graph[MAX]<\/code>: Adjacency list representation of the graph.<\/li>\ncolor[MAX]<\/code>: An array representing the color of each vertex.<\/li>\nqueue<int> q<\/code>: The queue used for BFS.<\/li>\n<\/ul>\nInput Example and Output Result<\/h3>\n
\nEnter the number of vertices and edges: 6 5\nInput edges (format: a b): \n1 2\n1 3\n2 4\n2 5\n3 6\n<\/pre>\n
\nThis graph is a bipartite graph.\n<\/pre>\n
Conclusion<\/h2>\n