Definition:Spanning Tree

Definition
Let $$G$$ be a connected graph.

A spanning tree for $$G$$ is a subgraph of $$G$$ which contains every vertex of $$G$$ and is also a tree.

Clearly a tree is its own spanning tree:

As a tree $$T$$ of order $n$ has $n-1$ edges, its spanning tree must also contain $$n-1$$ edges, and those must be the same ones as in $$T$$.

Creation of a Spanning Tree
There are two ways of creating a spanning tree for a given graph $$G$$:

Building-Up Method
Start with the null graph $$N$$ whose vertices correspond with those of $$G$$.

Select edges of $$G$$ one by one, such that no cycles are created, and add them to $$N$$.

Continue till all vertices are included.

Cutting-Down Method
Start with the graph $$G$$.

Choose any cycle in $$G$$, and remove any one of its edges.

By Condition for an Edge to be a Bridge, this will not disconnect $$G$$.

Repeat this procedure till no cycles are left in $$G$$.