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Chapter 11: Q36SE (page 805)

Which of the graphs in Exercise 35 are caterpillars?

Short Answer

Expert verified

(a): Therefore, the given tree is caterpillar. The graph of the tree is shown below.

(b): Therefore, the given tree is caterpillar. The graph of the tree is shown below.

(c): Therefore, the given tree is caterpillar. The graph of the tree is shown below.

(d): Therefore, the given tree is not a caterpillar. The graph of the tree is shown

Step by step solution

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01

General form

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

Definition of Circuit: It is a path that begins and ends in the same vertex.

Definition of Caterpillar: A caterpillar is a tree that contains a simple path such that every vertex not contained in this path is adjacent to a vertex in the path.

02

Proof of the given graphs are caterpillar or not

Referring from Exercise 35: the graphs are,

(a)

Given that, a graph of the tree.

For example, let the path be the edges that form a horizontal line in the graph. And we noticed that all vertices are in the path and so all vertices not in the path are adjacent to a vertex in the path. Graph is shown below.

Conclusion: So, the given graph is caterpillar.

(b)

Given that, a graph of the tree.

For example, let the path be the edges that form a horizontal line in the graph. And we noticed that all vertices are in the path and so all vertices not in the path are adjacent to a vertex in the path. Graph is shown below.

Conclusion: So, the given graph is caterpillar.

03

Proof of the given graphs are graceful or not

(c)

Given that, a graph of the tree.

For example, let the path be the edges that form a horizontal line in the graph. And we noticed that all vertices are in the path and so all vertices not in the path are adjacent to a vertex in the path. Graph is shown below.

Conclusion: So, the given graph is caterpillar.

(d)

Given that, a graph of the tree.

For example, let the path be the edges that form a horizontal line in the graph. And we noticed that all vertices are in the path and so all vertices not in the path are adjacent to a vertex in the path.

Since the given graph is not a caterpillar, because a path has at most length 4 in the graph. Here, we notice that one of the remaining vertices not in the path is adjacent to one of the vertices in the path, while the other vertex is not adjacent to any vertex in the path.

The graph is shown below.

Conclusion: So, the given graph is not a caterpillar.

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Most popular questions from this chapter

Show that a directed graph \({\bf{G = }}\left( {{\bf{V,E}}} \right)\) has an arborescence rooted at the vertex r if and only if for every vertex \({\bf{v}} \in {\bf{V}}\), there is a directed path from r to v.

Show that if \(G\) is a weighted graph with distinct edgeweights, then for every simple circuit of \(G\), the edge of maximum weight in this circuit does not belong to anyminimum spanning tree of \(G\).

In this exercise we will develop an algorithm to find the strong components of a directed graph \({\bf{G = }}\left( {{\bf{V,E}}} \right)\). Recall that a vertex \({\bf{w}} \in {\bf{V}}\) is reachable from a vertex \({\bf{v}} \in {\bf{V}}\) if there is a directed path from v to w.

  1. Explain how to use breadth-first search in the directed graph G to find all the vertices reachable from a vertex \({\bf{v}} \in {\bf{G}}\).
  2. Explain how to use breadth-first search in \({{\bf{G}}^{{\bf{conv}}}}\) to find all the vertices from which a vertex \({\bf{v}} \in {\bf{G}}\) is reachable. (Recall that \({{\bf{G}}^{{\bf{conv}}}}\) is the directed graph obtained from G by reversing the direction of all its edges.)
  3. Explain how to use part (a) and (b) to construct an algorithm that finds the strong components of a directed graph G, and explain why your algorithm is correct.

Show that the average depth of a leaf in a binary tree with \(n\) vertices is \({\bf{\Omega (logn)}}\).

How many vertices, leaves, and internal vertices does the rooted Fibonacci tree \({T_n}\) have, where \(n\) is a positive integer? What is its height?
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