find the graph is minimal spanig tree or not

To determine if a graph is a minimal spanning tree (MST) or not, you can use algorithms such as Kruskal's or Prim's algorithm. These algorithms help find the minimum spanning tree of a connected, weighted graph.

Kruskal's Algorithm:

  1. Sort all the edges of the graph in non-decreasing order of their weights.
  2. Initialize an empty tree to store the MST.
  3. Iterate through the sorted edges in the order of their weights.
  4. For each edge, check if adding it to the MST forms a cycle. If not, add the edge to the MST.
  5. Continue this process until all vertices are included in the MST or there are no more edges left.

Prim's Algorithm:

  1. Choose a starting vertex.
  2. Initialize an empty tree to store the MST.
  3. Create a set to store the vertices that have been included in the MST.
  4. Assign a weight of infinity to all vertices except the starting vertex (assign 0 to the starting vertex).
  5. While the set of included vertices is not equal to all the vertices:
  6. Select the vertex with the minimum weight that is not included in the MST.
  7. Add this vertex to the MST and update the weights of its adjacent vertices.
  8. Repeat this step until all vertices are included in the MST.

Both algorithms ensure that the resulting tree is a minimum spanning tree by adding edges with the minimum weight that do not form cycles. If the resulting tree connects all vertices of the original graph and has the minimum total weight, then it is a minimal spanning tree.

You can implement either of these algorithms in C++ to find the minimal spanning tree of a given graph.