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Chapter 12: Problem 7

Give an example of an undirected graph \(G=(V, E)\) where \(|V|=|E|+1\) but \(G\) is not a tree.

Short Answer

Expert verified
One example of an undirected graph \(G=(V, E)\) for which \(|V|=|E|+1\) but \(G\) is not a tree is the graph with 4 vertices A, B, C, D, and edges AB, BC, CA. It forms a cycle ABC, with D connected to A only.

Step by step solution

01

Construct an undirected graph

Firstly, let's construct a simple graph that fulfills the condition \(|V|=|E|+1\) but is not a tree. We could start by creating three vertices A, B, and C and connect them in a circular manner that forms a cycle. We then add an extra vertex D that is only connected to one of these vertices, for example, A. This graph is not a tree because it has a cycle.
02

Analyzing the graph

This graph \(G=(V, E)\) has 4 vertices \(V=\{A, B, C, D\}\) and 3 edges \(E=\{AB, BC, CA\}\). It's seen that \(|V|=|E|+1\), yet it contains a cycle (ABC), which means that it is not a tree.

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Most popular questions from this chapter

Let \(T=(V, E)\) be a rooted tree with root \(r\). Define the relation \(\$$ on \)V\( by \)x \mathscr{R} y\(, for \)x, y \in V\(, if \)x=y\( or if \)x\( is on the path from \)r\( to \)y\(. Prove that \)\mathscr{R}$ is a partial order.

Let \(G=(V, E)\) be a loop-free connected undirected graph with \(v \in V\). a) Prove that \(\overline{G-v}=\bar{G}-v\). b) If \(v\) is an articulation point of \(G\), prove that \(v\) cannot be an articulation point of \(\bar{G}\).

Let \(G=(V, E)\) be a loop-free undirected graph. Define the relation \(9 R\) on \(E\) as follows: If \(e_{1}, e_{2} \in E\), then \(e_{1} \Re e_{2}\) if \(e_{1}=e_{2}\) or if \(e_{1}\) and \(e_{2}\) are edges of a cycle \(C\) in \(G\). a) Verify that \(\mathscr{\text { is an equivalence relation on }} E\). b) Describe the partition of \(E\) induced by 9 .

Construct an optimal prefix code for the symbols \(a, b, c, \ldots, l, j\) that occur (in a given sample) with respective frequencies \(78,16,30,35,125,31,20,50,80,3 .\)

If \(B_{1}, B_{2}, \ldots, B_{k}\) are the biconnected components of a loop-free connected undirected graph \(G\), how is \(\chi(G)\) related to \(\chi\left(B_{i}\right), 1 \leq i \leq k\) ?
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