Section 5.6 Trees
Subsection 5.6.1 Terminology
Definition 5.6.1. Tree.
A graph is a tree if and only if it is connected and contains no cycles.
Definition 5.6.2. Leaf.
A vertex in a tree is a leaf if and only if it has degree one.
Definition 5.6.3. Spanning Tree.
A graph \(G\) is a spanning tree of a graph H if and only if \(G\) is a subgraph of \(H\) that contains all the vertices of \(H\) and is a tree.
Subsection 5.6.2 Practice
Checkpoint 5.6.4.
Determine which graphs in Figure Figure 5.1.1 are trees.
Checkpoint 5.6.5.
Draw all non-isomorphic trees on 3 vertices.
Checkpoint 5.6.6.
Prove that every tree with at least two vertices has at least one leaf.
Checkpoint 5.6.7.
Draw a tree with 7 vertices. Determine for what \(n\) the tree is \(n\)-connected.
Checkpoint 5.6.8.
Explain why for any tree removal of a leaf produces another tree.
Checkpoint 5.6.9.
Find a spanning tree for every graph in Figure 5.2.43.