Question

Let \(G=(V, E)\) be a graph. Prove that if \(|E|<|V|-1\), then \(G\) is disconnected.

Short answer

If \(|E| < |V| - 1\), \(G\) is disconnected.

Step-by-step solution

Understand the Problem

We are given a graph \(G=(V, E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. We are to prove that if the number of edges is less than the number of vertices minus one, \(|E| < |V| - 1\), then the graph \(G\) is disconnected.

Review Connected Graph Conditions

A connected graph with \(n\) vertices (\(|V|=n\)) must have at least \(n-1\) edges because a minimum spanning tree, which is a connected structure with no cycles, has exactly \(n-1\) edges.

Analyze Condition \(|E| < |V| - 1\)

If \(|E| < |V| - 1\), it means that the graph has fewer edges than a spanning tree would. This implies there aren't enough edges to connect all vertices, meaning it must be disconnected.

Proof by Contradiction Setup

Assume, for contradiction, that \(G\) is connected. By definition, a connected graph with \(n\) vertices must have at least \(n-1\) edges, but we have \(|E| < |V| - 1\). This is a contradiction.

Conclusion

Since assuming the graph is connected leads to a contradiction, the graph \(G\) must be disconnected if \(|E| < |V| - 1\).

Key concepts

Connected Graphs

In graph theory, a connected graph is a fundamental concept. Imagine a network of towns connected by roads. A connected graph means there's at least one way to travel between any two towns in this network without leaving the roads. In simpler terms, a connected graph has pathways connecting all its vertices (or nodes). This ensures there is no isolated vertex.

• Each node can reach any other node via a series of edges.


• The graph has no separate parts.


• If you start anywhere, you can reach every other node.

To maintain these connections in a graph with 'n' vertices, at least \(n - 1\) edges are required. This number ensures there are enough pathways to bridge all vertices directly or indirectly without redundancies.

Minimum Spanning Tree

A minimum spanning tree (MST) is an essential structure in a connected graph. Picture it as the minimum number of roads needed to connect all towns without any extra detours or loops. It's the cheapest way to connect all points in a network if each road has a cost.

• It includes all the vertices but uses the least number of edges.


• Has exactly \(n-1\) edges for a graph with \(n\) vertices.


• It forms a tree structure, meaning no cycles.

In real life, the concept of an MST helps design efficient networks such as utilities, communication lines, and road systems, by minimizing construction costs while maintaining full connectivity.

Disconnected Graphs

A disconnected graph is the opposite of a connected graph. If we imagine a network again, some towns have no roads connecting them to this network or isolated groups of towns. In technical terms, not every pair of vertices can be connected by a sequence of edges.

• There are separate clusters of connected nodes.


• Some vertices might be completely isolated from others.


• It's possible for parts of the graph to have no connection between them.

If a graph has fewer edges than required for a minimum spanning tree, \(|E| < |V| - 1\), it means the graph doesn't have enough bridges to maintain connectivity across all vertices, leading to disconnection. This is crucial to understanding when designing networks because it shows the minimum resources needed to maintain a completely interconnected system.