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).

Template:ExerciseRobox 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.

$tree$

Acyclic and connected $graph$.
$forest$ is acyclic graph. So, $tree\subseteq forest$

�Significance

• tree : minimum size connected graph
• cycle : minimum size 2-connected graph