Graph Theory Questions Medium
A minimum spanning tree (MST) is a tree that spans all the vertices of a connected, weighted graph with the minimum possible total weight. In other words, it is a subset of the edges of the graph that connects all the vertices together without forming any cycles and has the minimum sum of edge weights.
To find a minimum spanning tree, various algorithms can be used, such as Kruskal's algorithm or Prim's algorithm. Kruskal's algorithm starts with an empty set of edges and iteratively adds the edges with the smallest weights that do not create a cycle until all vertices are connected. Prim's algorithm starts with a single vertex and grows the tree by adding the edge with the smallest weight that connects a vertex in the tree to a vertex outside the tree until all vertices are included.
The minimum spanning tree has several applications, including network design, clustering, and optimization problems. It ensures that the graph is connected while minimizing the total cost or weight required to connect all the vertices.