Question

Suppose that a graph \(G\) is connected, but, for each edge, deleting that edge leaves a disconnected graph. What can you say about \(G\) ? Prove it.

Short answer

The graph is a tree.

Step-by-step solution

Understanding the Problem

We are given a connected graph where deleting any edge results in a disconnected graph. We must determine what type of graph has this property and prove it.

Define a Bridge

A bridge in a graph is an edge that, when removed, increases the number of connected components of the graph. In this scenario, each edge has this property, as the removal of any edge disconnects the graph.

Identify the Graph Type

The graph we are describing is a tree. A tree is a special case of a connected graph where there is exactly one path between any two vertices, and every edge is a bridge.

Prove that the Graph is a Tree

1. **Connectedness**: By definition, the graph is connected. 2. **Acyclic**: Assume, for contradiction, that there is a cycle in the graph. Removing an edge from a cycle leaves the graph connected, contradicting that removing any edge disconnects the graph. 3. Therefore, the graph has no cycles (is acyclic) and is connected.

Conclusion

Since the graph is connected and acyclic, it satisfies the definition of a tree. Each edge is a bridge, meaning removing any edge disconnects the graph, confirming our initial observation.

Key concepts

Connected Graph

A connected graph is one where there is a path between every pair of vertices. This is an essential property in graph theory, ensuring that no vertex is isolated from the others. In such a graph, if you start at any vertex, you can traverse through edges to reach any other vertex.
This property makes connected graphs particularly useful for modeling networks where it's crucial for all points to remain reachable to each other. For example, consider a city map: each intersection is a vertex, and roads are edges. A connected city map ensures you can travel from any intersection to any other without leaving the main roads.
In simpler terms, a connected graph behaves like a unified system. If a graph is disconnected, it consists of multiple components, where vertices within a component are connected, but there's no path from one component to any vertex in another component.

Tree

In graph theory, a tree is a special type of connected graph with no cycles. This means that there's exactly one unique path connecting any pair of vertices. Some fundamental properties of trees include:

• They are connected: As already noted, every pair of vertices has one path joining them.
• They are acyclic: This means trees don't have cycles. Imagine removing an edge from a cycle—the group would still be connected, contradicting an essential tree property.
• Edges as bridges: Every edge in a tree acts as a bridge. Removing any edge from a tree will disconnect the tree.
• Vertex and edge relationship: For a tree with \( n \) vertices, there are precisely \( n-1 \) edges.

Because of their unique properties, trees are often used in computer science, such as in organizing hierarchical data structures and designing efficient algorithms.

Bridge in Graph

An edge in a graph is referred to as a bridge if its removal increases the number of connected components in the graph. In simpler terms, eliminating a bridge creates a divide in the graph, disconnecting it.
Bridges are critical in network design because they identify points of vulnerability. If a network’s bridge fails, the network could split into isolated parts, losing overall connectivity.
In a tree, every edge serves as a bridge, meaning that removing any edge results in a disconnected graph. This is what differentiates a tree from other connected graphs. It's crucial in networks, as it helps in maintaining integrity and redundancy by ensuring that no single failure can completely split the network.