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Chapter 11: Problem 10

Give an example of a connected graph \(G\) where removing any edge of \(G\) results in a disconnected graph.

Short Answer

Expert verified
An example of a connected graph that becomes disconnected with the removal of any edge is a graph with three vertices (A, B, and C) and two edges (AB and BC).

Step by step solution

01

Definition of a Connected Graph

A connected graph is a type of graph where a path exists between every pair of vertices. This means that, starting from any vertex, we can reach any other vertex in the graph.
02

Create a Bridge Graph

Based on the conditions provided in the exercise, we can create a graph with three vertices, say, \(A\), \(B\), and \(C\), and two edges, say, \(AB\) and \(BC\). These two edges form a bridge between \(A\) and \(C\). This graph (ABC) is an example of a connected graph.
03

Graph Testing

Now, let us remove an edge from the graph, say, \(AB\). After removing \(AB\), there is no direct path left between \(A\) and \(B\) or from \(A\) to \(C\), and so the graph becomes disconnected. Similarly, if we remove \(BC\), there is no path from \(B\) or \(C\) to \(A\), making the graph disconnected again. This confirms that our example meets the conditions of the exercise.

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Most popular questions from this chapter

Prove that for any \(n \in \mathbf{Z}^{+}\)there exists a loop-free connected undirected graph \(G=\) \((V, E)\) where \(|V|=2 n\) and which has two vertices of degree \(i\) for every \(1 \leq i \leq n\).

Let \(G\) be a loop-free undirected graph where \(\Delta=\max _{v e v}\\{\) deg \((v)\\}\). a) Prove that \(\chi(G) \leq \Delta+1\). b) Find two types of graphs \(G\) where \(\chi(G)=\Delta+1\).

a) Let \(V=\\{a, b, c, d, e, f\\} .\) Draw three nonisomorphic loop-free undirected graphs \(G_{1}=\) \(\left(V, E_{1}\right), G_{2}=\left(V, E_{2}\right)\), and \(G_{3}=\left(V, E_{3}\right)\) where, in all three graphs, we have \(\operatorname{deg}(a)=\) \(3, \operatorname{deg}(b)=\operatorname{deg}(c)=2\), and \(\operatorname{deg}(d)=\operatorname{deg}(e)=\operatorname{deg}(f)=1 .\) b) How many of the graphs in part (a) are connected?

a) Find the maximum length of a trail in i) \(K_{6}\) ii) \(K_{\mathrm{s}}\) iii) \(K_{10}\) iv) \(K_{2 n}, n \in \mathbf{Z}^{+}\). b) Find the maximum length of a circuit in i) \(K_{6}\) ii) \(K_{8}\) iii) \(K_{10}\) iv) \(K_{2 n}, n \in \mathbf{Z}^{+}\).

Let \(G=(V, E)\) be an undirected graph, where \(|V| \geq 2\). If every induced subgraph of \(G\) is connected, can we identify the graph \(G ?\)
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