Question

Find a graph \(G\) such that \(\bar{G}\) is not connected.

Short answer

Use a complete graph on two vertices \(K_2\). \(\bar{G}\) will be disconnected.

Step-by-step solution

Understanding the Complement of a Graph

To solve this problem, we need to understand the concept of a graph's complement. For a given graph \(G = (V, E)\), the complement graph \(\bar{G}\) is a graph on the same vertices \(V\). Two vertices \(u\) and \(v\) are connected by an edge in \(\bar{G}\) if and only if they are not connected by an edge in \(G\).

Consider a Graph with Few or No Edges

To find a graph \(G\) such that \(\bar{G}\) is not connected, we should consider a graph \(G\) with few or very specific connections among vertices. The simplest scenario is when \(G\) has few edges, creating isolated components in \(\bar{G}\).

Construct a Specific Graph

One simple example is to take \(G\) as a complete graph on two vertices \(K_2\). In this case, \(G\) is a single edge connecting two vertices. The complement \(\bar{G}\) would have these two vertices but with no edge connecting them, making \(\bar{G}\) a graph with two disconnected vertices.

Key concepts

Graph Complement

When we talk about the graph complement, we refer to a transformation that occurs to a graph's edges while keeping its vertices intact. Imagine a graph \( G = (V, E) \), where \( V \) is a set of vertices and \( E \) is a set of edges connecting some pairs of these vertices. The complement of this graph, denoted \( \bar{G} \), uses the same set of vertices \( V \), but the edges are different.
In \( \bar{G} \), we only connect vertices with an edge if there isn't one between the same pair in the original graph \( G \). It essentially inverts the presence of edges: if vertices had an edge in \( G \), they don't in \( \bar{G} \), and if they didn't in \( G \), they do in \( \bar{G} \).
This property makes the graph complement concept quite intriguing in graph theory. The defining rule of edge existence flips, offering a new perspective on the relationship between different graph structures.
The exercise highlights the complement of a graph on a simple level. For instance, by taking a complete graph of two vertices \( K_2 \), \( G \) would consist of just one edge joining them. Therein, its complement \( \bar{G} \), would display those vertices without an edge, rendering \( \bar{G} \) completely unconnected.

Discrete Mathematics

Discrete mathematics encompasses the study of mathematical structures that are fundamentally discrete, meaning they do not require the concept of continuity. At its core, this area of mathematics deals with objects such as integers, graphs, and predicates, differing from continuous mathematics, which involves calculus and real analysis.
Discrete mathematics provides the foundations for graph theory, which examines structures made up of nodes and their pairwise connections, known as edges. Such frameworks are pivotal in computer science, particularly in algorithms, data structures, and coding theory.
Why is discrete mathematics essential to graph theory? Simply put, because graphs are a collection of discrete elements—vertices and edges—connected in specific ways. This results in systems able to model networks, from social connections to information flow patterns and computational processes.

• Graphs and graph complements are core components of discrete mathematics.
• Problem-solving often involves considering many structural variations within graphs, including their connectedness.
• Understanding discrete mathematical principles aids greatly in unraveling many graph theoretical issues.

This rich discipline provides tools to navigate complex systems discretely, often yielding insights applicable to tangible problems.

Connected Graphs

A connected graph, in simple terms, is one in which there is a path between every pair of vertices. This ensures that no vertex remains isolated from another. It’s like ensuring each person at a party can reach every other person, perhaps indirectly, via a series of handshake-trails.
When we talk about the graph complement in the context of connectivity, things get fascinating. The complement \( \bar{G} \) of a graph \( G \) could end up being disconnected. This happens when the inverse relationship of edge presence leads to sections of the graph no longer connected.
For example, consider a basic graph and its complement. Take \( G \) as having merely one connection, say a single edge between two vertices. In this configuration, the complement graph \( \bar{G} \) would depict these vertices without connection, symbolizing two isolated points. Such graphs demand careful consideration because they illustrate how altering connectivity impacts graph characteristics.

• A connected graph ensures continuous paths throughout all vertices within the graph.
• Graph complements can highlight the absence of connectivity with the removal of essential links.
• This exercise challenges understanding, showing how slight changes in graph structure can lead to drastic differences in their complement.

Connected graphs are foundational in studying and modeling networks, making their understanding key across multiple disciplines.