(a)
To define:
A tree.
Explanation of Solution
Definition: A tree is a connected undirected graph with no simple circuits. Because a tree cannot have a simple circuit, a tree cannot contain multiple edges or loops. Therefore any tree must be a simple graph.
An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.
Example:
(b)
To define:
A forest.
Explanation of Solution
Definition: A forest is an undirected graph, all of whose connected components are trees.
i.e. the graph consists of a disjoint union of trees.
A forest is a disjoint union of trees, or equivalently anacyclic graph that is not necessarily connected.
Example:
An empty graph, a single tree and the discrete graph on a set of vertices are examples of forest tree.
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Chapter 11 Solutions
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