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Chapter 12: Problem 34

Prove that the complement of a disconnected graph is connected.

Short Answer

Expert verified
In the complement graph Gc for a given disconnected graph G, a connection always exists between any given pair of vertices. This can be because they were separated in G (in different subsets V1, V2) or because there was no connecting edge between them in the same subset of G. Therefore, the complement of a disconnected graph is always connected.

Step by step solution

01

Define disconnectivity

Let's first define the properties of a disconnected graph. A graph G is said to be disconnected if there exist two vertices in G such that no path in G has these vertices as endpoints. This means, if we consider the total set of vertices V in G, it can be partitioned into two non-empty subsets V1 and V2 so that no edge connects a vertex in V1 with a vertex in V2.
02

Define graph complement

The complement graph denoted as Gc of a graph G is formed by using the same vertices as G, but the edge set of Gc is the complement to the edge set of G with respect to the complete graph. This means every edge of Gc connects two vertices that are not connected in G.
03

Prove connection in Gc

Now, given that G is disconnected, we know it can be partitioned into two non-empty sets V1 and V2 with no connecting edge. Now pick any two vertices u and v in Gc, if they are both in V1 or in V2, there is an edge between them in Gc because there is no such edge in G. If one of them is in V1 and the other is in V2, again there exists an edge in Gc, since V1 and V2 are disconnected in G. Therefore, in Gc, there exists a path that includes any pair of vertices which means Gc is connected.

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Key Concepts

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

Disconnected Graph
In graph theory, a disconnected graph is a type of graph where not all vertices are linked together through paths. Imagine a group of islands where no bridge connects some islands to others. That is akin to a disconnected graph. In technical terms, a graph \( G \) is disconnected when there are at least two vertices with no path between them.
  • A disconnected graph can be split into multiple subsets.
  • Each subset is called a component, and each component is a smaller graph partially disconnected from the rest.
  • These components have vertices that are only interconnected within themselves and not with any other component.
When considering a disconnected graph \( G \), dividing it into parts simplifies understanding. We view it as a family of smaller, isolated networks.
Graph Complement
To understand the complement of a graph \( G \), we transform \( G \) into another graph called \( G^c \). The twist is, \( G^c \) includes every possible edge that was missing from \( G \).
  • The vertices of \( G^c \) are the same as the vertices of \( G \).
  • If two vertices in \( G \) were not linked, in \( G^c \), they are connected.
  • Conversely, if two vertices were connected in \( G \), then in \( G^c \), they are not connected.
To picture this, think of a social network where you compile a new database that connects everyone who wasn't already friends. The essence of graph complement is about connecting those who were previously unconnected.
Connected Graph
A graph is termed as 'connected' when you can reach any vertex from any other vertex through a sequence of edges. Imagine a bustling city where each neighborhood is connected to every other one by roads; that city layout forms what we call a connected graph. In more formal terms, a graph \( G \) is connected if there exists a path between every pair of distinct vertices.
  • In a connected graph, there are no isolated parts.
  • For every vertex, there is at least one edge linking it to the rest of the graph.
  • If you were to remove any single edge from a connected graph, the graph might become disconnected.
Remember, the complement of a disconnected graph tends to be connected based on how it's reformed, like merging isolated groups through new roads ensuring unity across the graph.

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Most popular questions from this chapter

Prove that all bipartite graphs are perfect.

Prove that the only connected bipartite graphs that are chordal are trees.

Prove that a connected graph can always be contracted to a single vertex.

Let \(G\) be a graph such that either \(G\) or its complement \(\bar{G}\) has an induced sub graph equal to a chordless cycle of odd length greater than 3. Prove that \(G\) is not perfect.

Let \(G\) be a graph each of whose vertices has positive degree. Prove that \(G\) is 2 . connected if and only if, for each pair of edges \(\alpha_{1}, \alpha_{2}\), there is a cycle containing both \(\alpha_{1}\) and \(\alpha_{2}\).
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