Check if there Exists a Connected Graph that Satisfies the Given Conditions

To determine if an associated chart exists that fulfils the given conditions, we can utilise a basic approach. The conditions state that the chart must have at least one hub with an odd degree and all other hubs must have an indeed degree. We are able to make such a chart by beginning with a single hub and steadily including hubs and interfacing them in sets. For each unused hub included, we interface it to an existing hub, guaranteeing that the existing hub has an indeed degree and the modern hub has an odd degree. By proceeding with this preparation until all hubs are associated, we are able to build an associated chart that fulfils the given conditions.

Methods Used

• Graph traversal approach

Graph Traversal Approach

To check on the off chance that an associated chart fulfils certain conditions, a common approach is to utilise chart traversal calculations like Depth−First Look (DFS) or Breadth−First Look (BFS). Beginning from any vertex, navigate the chart by going to its neighbouring vertices until all vertices are gone or the conditions are damaged. On the off chance that all conditions are met and all vertices are gone, at that point an associated chart fulfilling the given conditions exists. The chart traversal calculations offer assistance in investigating the network of the chart and guarantee that all vertices are reachable from a beginning point.

Algorithm

• Initialise a boolean cluster, gone to[], to keep track of gone−by vertices.

• Choose a beginning vertex and stamp it as visited.

• Create a purge stack or line information structure.

• Push or enqueue the beginning vertex into the stack or queue.

• While the stack or queue isn't purgeable, do the following:

• a. Pop or dequeue a vertex from the stack or queue.

• b. Prepare the vertex according to the given conditions.

• c. Stamp the vertex as visited.

• d. Push or enqueue all adjoining unvisited vertices into the stack or queue.

• After navigating the complete chart, check to see if all vertices have been visited.

• If all vertices are gone and the conditions are fulfilled, yield "Associated chart exists."

• Otherwise, yield "No associated chart exists."

Example

#include <iostream>
#include <vector>

using namespace std;

// Function to perform DFS traversal
void DFS(vector<vector<int>>& graph, vector<bool>& visited, int vertex) {
    visited[vertex] = true;

    // Traverse all adjacent vertices
    for (int neighbor : graph[vertex]) {
        if (!visited[neighbor]) {
            DFS(graph, visited, neighbor);
        }
    }
}

// Function to check if a connected graph satisfying conditions exists
bool isConnectedGraph(vector<vector<int>>& graph) {
    int numVertices = graph.size();

    // Mark all vertices as not visited
    vector<bool> visited(numVertices, false);

    // Find the first non-empty vertex and perform DFS
    int startVertex = -1;
    for (int i = 0; i < numVertices; ++i) {
        if (!graph[i].empty()) {
            startVertex = i;
            break;
        }
    }

    if (startVertex == -1) {
        // Empty graph, no conditions can be satisfied
        return false;
    }

    // Perform DFS traversal from the start vertex
    DFS(graph, visited, startVertex);

    // Check if all vertices are visited
    for (bool v : visited) {
        if (!v) {
            // Graph is not connected, conditions cannot be satisfied
            return false;
        }
    }

    // All vertices are visited, the graph is connected and satisfies conditions
    return true;
}

int main() {
    // Create the graph adjacency list
    vector<vector<int>> graph = {
        {1, 2},       // Vertex 0 is connected to vertices 1 and 2
        {0, 2},       // Vertex 1 is connected to vertices 0 and 2
        {0, 1, 3},    // Vertex 2 is connected to vertices 0, 1, and 3
        {2}           // Vertex 3 is connected to vertex 2
    };

    // Check if a connected graph satisfying conditions exists
    bool isConnected = isConnectedGraph(graph);

    // Output the result
    if (isConnected) {
        cout << "A connected graph satisfying the conditions exists." << endl;
    } else {
        cout << "No connected graph satisfying the conditions exists." << endl;
    }

    return 0;
}

Output

A connected graph satisfying the conditions exists.

Conclusion

This article examines the issue of determining the presence of an associated chart that fulfils given conditions. It investigates the concept of developing a chart methodically to meet particular prerequisites. The article clarifies the algorithmic approach of including vertices and interfacing them with edges while guaranteeing a network. It gives a step−by−step procedure to form a chart that fulfils the given conditions. The article, too, incorporates a code scrap in C to illustrate the execution of the chart development calculation. By and large, it offers bits of knowledge for fathoming the issue of finding an associated chart that meets the indicated criteria.