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Chapter 11: Q1E (page 755)

Which of these graphs are trees?

(a)

(b)

(c)

(d)

(e)

(f)

Short Answer

Expert verified
  1. Is a tree.
  2. Is not a tree.
  3. Is a tree.
  4. Is not a tree.
  5. Is a tree.
  6. Is not a tree.

Step by step solution

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01

Solution

A tree is a connected undirected graph with no simple circuits.

02

(a)

This is a tree.

03

(b)

This is not a tree because it is not connected.

04

(c)

This is a tree.

05

(d)

This is not a tree because it has a simple circuit.

06

(e)

This is a tree.

07

(f)

This is not a tree because it is a circuit.

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Most popular questions from this chapter

a) What is the height of a rooted tree?

b) What is a balanced tree?

c) How many leaves can an \({\bf{m}}\)-ary tree of height \({\bf{h}}\) have?

Show that a simple graph is a tree if and only if it contains no simple circuits and the addition of an edge connecting two nonadjacent vertices produces a new graph that has exactly one simple circuit (where circuits that contain the same edges are not considered different).

Find the second least expensive communications network connecting the five computer centers in the problem posed at the beginning of the section.

The eccentricity of a vertex in an unrooted tree is the length of the longest simple path beginning at this vertex. A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex. In Exercises \({\bf{39--41}}\) find every vertex that is a center in the given tree.

41.

a. What is a minimum spanning tree of a connected weighted graph\(?\)

b. Describe at least two different applications that require that a minimum spanning tree of a connected weighted graph be found.
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