Explanation:

To understand why the correct answer is option 'D' (tree), let's break down the characteristics of each option:

a) Hamiltonian cycle: A Hamiltonian cycle is a cycle in a graph that visits every vertex exactly once. While it does connect all the vertices, it is not guaranteed to have the minimum total weight. The weights of the edges can vary, and there is no guarantee that a Hamiltonian cycle will have the lowest total weight.

b) Grid: A grid is a graph formed by connecting vertices in a rectangular grid pattern. While it is possible to connect all the vertices in a grid, it is not necessary to use all the edges to achieve this. Additionally, the total weight of the edges in a grid can vary, so it is not necessarily the minimum.

c) Hypercube: A hypercube is a graph formed by connecting vertices in an n-dimensional cube. Similar to a grid, it is possible to connect all the vertices in a hypercube, but it is not necessary to use all the edges to achieve this. Also, the total weight of the edges can vary, so it is not necessarily the minimum.

d) Tree: A tree is a connected acyclic graph. In a tree, there is exactly one path between any two vertices, and all the vertices are connected. If all the edge weights of an undirected graph are positive, the minimum total weight subset of edges that connects all the vertices will form a tree. This is known as a minimum spanning tree (MST).

The reason a tree is the correct answer is because of the properties of an MST:

1. A tree connects all the vertices: In an MST, all the vertices of the graph are connected, ensuring that every vertex can be reached from any other vertex.

2. The total weight is minimized: An MST has the minimum total weight among all possible subsets of edges that connect all the vertices. This is because the algorithm for finding an MST (such as Prim's algorithm or Kruskal's algorithm) carefully selects edges with the minimum weight while ensuring that no cycles are formed.

So, in conclusion, if all the edge weights of an undirected graph are positive, the subset of edges that connects all the vertices and has the minimum total weight will form a tree (option 'D').