Graph Theory/Trees

A tree is a type of connected graph. An directed graph is a tree if it is connected, has no cycles and all vertices have at most one parent. An undirected graph is considered a tree if it is connected, has $$|V| - 1$$ edges and is acyclic (a graph that satisfies any two of these properties satisfies all three).

Show that the following are equivalent definitions for a tree:


 * A graph with a minimal number of edges which is connected.
 * A graph with maximal number of edges without a cycle.
 * A graph with no cycle in which adding any edge creates a cycle.
 * A graph with n nodes and n-1 edges that is connected.
 * A graph in which any two nodes are connected by a unique path (path edges may only be traversed once).

Hint: To keep the total proof short, put the definitions in a suitable order, and then prove A=>B=>C=>D=>E=>A. Take particular care over graphs with zero and one node. Additionally,
 * A graph is connected and each edge is a bridge.

Acyclic and connected $$graph$$. $$forest$$ is acyclic graph. So, $$tree \subseteq forest$$
 * $$tree$$
 * Meaning
 * tree : minimum connected graph
 * cycle : minimum 2-connected graph