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Chapter 2: Problem 5

Prove that a simple graph and its complement cannot both be disconnected.

Short Answer

Expert verified
No, a simple graph and its complement cannot both be disconnected. If you assume the contrary, you would reach a contradiction. If the graph is disconnected, it has at least two components. And in the complement, each vertex of one component would be connected to every vertex of the other, making it a connected graph. This contradicts the initial assumption that both the graph and its complement are disconnected.

Step by step solution

01

Assume the contrary

In order to prove that a simple graph and its complement cannot both be disconnected, we assume the contrary i.e., we assume there exists a simple graph and its complement that are both disconnected. Now, try to find a contradiction to invalidate our assumption.
02

Analyze the simple graph

Since the graph is assumed to be disconnected, there exist at least two components, let them be A and B. If A and B are distinct connected components of the graph then there does not exist an edge between any vertex of A and any vertex of B.
03

Analyze its complement

In the complement of this simple graph, any two vertices that were not adjacent in the initial graph are now adjacent. That means, in the complement of our graph every vertex of A will be connected to every vertex of B, and every vertex of B will be connected to every vertex of A.
04

Find the contradiction

Having shown that every vertex in A is connected to every vertex in B in the complement graph contradicts the assumption that the complement graph is disconnected. Thus, the initial assumption that there exists a simple graph and its complement that are both disconnected is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Graphs
A simple graph is a fundamental concept in graph theory. It consists of a set of vertices and a set of edges. Each edge connects two distinct vertices, and no two edges connect the same pair of vertices. Simple graphs are significant because they do not have loops where an edge connects a vertex to itself, nor do they have multiple edges between the same pair of vertices.
A simple graph can be easily visualized as a network with points (vertices) connected by lines (edges). They serve as the basis for many problems and concepts in graph theory.
  • Vertices: The points in the graph.
  • Edges: The lines that connect the vertices.
  • No Loops or Multiple Edges: Simplifies the study of graph properties.
Understanding these basic components is crucial in exploring more complex ideas such as complement graphs and connected components.
Complement Graphs
The complement graph of a given simple graph is derived by swapping which vertex pairs are connected and which are not. For every two vertices in the original simple graph, if there is an edge between them, then in the complement graph, there won't be one. Conversely, if there is no edge between a pair of vertices in the original graph, there will be an edge in the complement graph.
  • Edge Switching: Edges exist in the complement graph where there were no edges before.
  • Complete Set of Edges: The graph becomes more connected where the original lacked connections.
  • Visualization: Imagine flipping all pairwise connections.
Understanding complement graphs helps solve problems involving connectivity, as seen in the exercise with disconnected and connected graphs. The complement graph provides insights into the inverse or opposing structures within graph theory.
Connected Components
In graph theory, connected components are subsets of a graph's vertices where there exists at least one path between any pair of vertices within the subset. If a graph is disconnected, it may have multiple such components. Each component represents a maximal set of connected vertices.
  • Connected vs. Disconnected: A graph is connected if there is a path connecting any two vertices.
  • Components: Each contains vertices that are mutually reachable.
  • Analysis: Examining components is critical for understanding the graph's structure.
Analyzing connected components is essential in understanding the nature of a graph, especially when reviewing transformations such as creating a complement graph. In typical scenarios, the transition from a disconnected simple graph to a complemented one often reveals unexpected connectivity, as shown in the exercise's analysis.

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Most popular questions from this chapter

The girth of a graph is the length of its shortest cycle. Write down the girths of the following graphs: (i) \(K_{9}\); (iii) \(C_{8}\); (iv) \(W_{8}\); (v) \(Q_{5}\); (vi) the Petersen graph; (vii) the graph of the dodecahedron.

A set \(E\) of edges of a graph \(G\) is independent if \(E\) contains no cycle of \(G\). Prove that: (i) any subset of an independent set is independent; (ii) if \(I\) and \(J\) are independent sets of edges with \(|J|>|I|\), then there is an edge \(e\) that lies in \(J\) but not in \(I\) with the property that \(I \cup\\{e\\}\) is independent. Prove also that (i) and (ii) still hold if we replace the word 'cycle' by 'cutset'.

Find orientations for the Petersen graph and the graph of the dodecahedron.

Let \(D\) be the digraph whose vertices are the pairs of integers \(11,12,13,21,22,23,31\), 32,33 , and whose arcs join \(i j\) to \(k l\) if and only if \(j=k\). Find an Eulerian trail in \(D\) and use it to obtain a circular arrangement of nine 1s, nine \(2 \mathrm{~s}\) and nine \(3 \mathrm{~s}\) in which each of the 27 possible triples (111, 233, etc.) occurs exactly once. (Problems of this kind arise in communication theory.)

(i) Prove that, if two distinct cycles of a graph \(G\) each contain an edge \(e\), then \(G\) has a cycle that does not contain \(e\). (ii) Prove a similar result with 'cycles' replaced by 'cutsets'.
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