Question

Prove that a graph \(T\) is a tree if and only if \(T\) is connected but the deletion of any edge disconnects \(T\).

Short answer

A graph \(T\) is a tree if and only if it is connected and edge deletion disconnects it.

Step-by-step solution

Understanding the Properties of a Tree

A tree is an undirected graph that is connected and contains no cycles. This means that there is exactly one path between any two vertices.

Prove the 'Only if' Direction

Assume that the graph \(T\) is a tree. By definition, \(T\) is connected. We need to show that if any edge is removed, the graph becomes disconnected. Since there is exactly one path between any two vertices in a tree, removing any edge destroys this unique path, thereby disconnecting the vertices that were previously connected directly. Therefore, removing any edge disconnects the tree.

Prove the 'If' Direction

Assume \(T\) is connected and removal of any edge disconnects \(T\). First, \(T\) is acyclic because having a cycle would mean removing one edge in the cycle will not disconnect the graph, contradicting our assumption. Second, since removal of any edge disconnects \(T\), this confirms that every edge in \(T\) is a bridge, establishing that there is a unique path between any two vertices, which implies the absence of cycles. Hence, \(T\) is a tree as it is both connected and acyclic.

Conclusion

Therefore, through this bi-directional proof, we have shown that a graph \(T\) is a tree if and only if it is connected and the deletion of any edge disconnects \(T\).

Key concepts

Tree

In graph theory, a **tree** is a special kind of graph. It is characterized by its structure that is both connected and acyclic. Let’s break this down:

• **Connected**: All vertices in the tree are connected in such a way that there is one and only one path between any pair of vertices.
• **Acyclic**: It contains no cycles. This means you cannot start at one vertex and return to it by traveling along a sequence of edges without retracing.

A practical example of trees in use is a family tree in genealogical charts or directory structures in computer science. They are straightforward yet incredibly useful, providing clarity in searching and organizing data.

Connected Graph

A **connected graph** is a graph in which there is a path between every pair of vertices. In other words, no vertex is isolated. If you pick any two vertices in a connected graph, you can find a route joining them via edges.

This is an important concept because it ensures that no part of the graph is unreachable. In the context of trees, connectedness ensures the cohesiveness of the graph, allowing each node to be part of a larger system. If a graph is not connected, it consists of two or more components, meaning it would not be a tree if it has more than one component.

Acyclic Graph

An **acyclic graph** is a graph that does not contain any cycle. A cycle is a path of edges and vertices wherein a vertex is reachable from itself.

• In trees, acyclicity is a defining property.
• If a graph is acyclic, like a tree, it eliminates the possibility of endless loops in the navigation of the graph.

Understanding acyclic graphs is crucial because cycles in graphs can complicate traversal processes. This becomes particularly beneficial in algorithms for detecting cycles or deadlocks in computational processes.

Edge Deletion

**Edge deletion** involves the removal of an edge or edges from a graph. In the context of trees, removing any edge will result in a disconnected graph because there is only one path between any two vertices.

Edge deletions are useful in graph algorithms, like when trying to simplify a structure or analyze connectivity. The act of deleting an edge in a tree underlines the essential connectivity trees provide, as every edge holds a critical role in maintaining the graph’s connectedness.

Bridges in Graphs

**Bridges in graphs** refer to edges whose removal increases the number of connected components of the graph. In simpler terms, a bridge is a critical connection in a graph.

• In a tree, every edge is a bridge. This is because removing any edge in a tree disconnects the graph into separate parts.
• Identifying bridges can help in understanding the robustness of a network because they represent points of failure that could disrupt the connectivity.

Bridges play a significant role in understanding connectivity resilience and the structural weaknesses within networks, making them crucial in network design and analysis.