Question

Let \(G\) be a connected graph. What can you say about (i) an edge of \(G\) that appears in every spanning tree? (ii) an edge of \(G\) that appears in no spanning tree?

Short answer

(i) An edge that appears in every spanning tree is a cut edge or bridge since it is crucial for maintaining the connectedness of the graph. (ii) An edge that appears in no spanning tree is a redundant edge or isthmus because it forms a cycle and excluding it doesn't disconnect the graph.

Step-by-step solution

Understand the concept of a cut edge

A cut edge or bridge is an edge in a connected graph that, if removed, increases the number of connected components. That means there would be vertices no longer reachable from other vertices. Therefore, a cut edge must appear in all spanning trees. Otherwise, if you remove it, you will have disconnected vertices not included in the spanning tree, which contradicts the definition of spanning tree.

Understand the concept of a redundant edge

A redundant edge or isthmus is an edge in a connected graph that forms a cycle and can be removed without disconnecting the graph. Meaning, if it's removed, all vertices remain reachable. Hence, this edge can't be part of a spanning tree because a spanning tree is acyclic and therefore mustn't include any cycles.

Key concepts

Cut Edge

In the world of graph theory, a **cut edge** (or bridge) is a pivotal concept. Imagine your graph as a network of cities connected by roads. A cut edge is like a major highway that, if taken out of service, would split the network into two, preventing some cities from being reachable from others. This demonstrates that a cut edge is crucial to maintaining the connectivity of the graph.

The defining characteristic of a cut edge is its necessity in all spanning trees of the graph. Spanning trees are subgraphs that include all vertices and are connected without forming cycles. Since the removal of a cut edge results in a disconnected graph, every spanning tree must incorporate this edge to stay connected. Without the cut edge, parts of the graph would become unreachable, contradicting the very essence of a spanning tree.

Understanding cut edges helps in analyzing the resilience of networks and ensuring that there is no single point of failure in a network system.

Spanning Tree

A **spanning tree** is an interesting element within graph theory because it offers a minimalistic view of connectivity. Think of a spanning tree as a blueprint for connecting all points (or vertices) in a network efficiently. This tree spans across all nodes ensuring each node is included and fully connected, but it does so without closing any loops, meaning it has no cycles.

Spanning trees are subgraphs that reflect the essential connections required to keep all vertices linked together. They capture the backbone structure of a graph, retaining connectivity using the smallest number of edges possible. As a result, a spanning tree for a graph with "n" vertices will always contain "n-1" edges, which is the absolute minimum needed to keep all vertices connected.

This concept is vital in applications like network design, where the goal is to ensure all nodes can communicate while minimizing the use of resources. Understanding spanning trees allows one to efficiently design pathways that prevent redundancy and maximize the use of available edge connections.

Redundant Edge

A **redundant edge** in graph theory adds an interesting twist to the connectivity of a graph. Imagine this as an extra road between cities that, while useful, isn’t necessary for reaching all destinations. Its removal wouldn’t split the network; every city remains connected. This is what makes a redundant edge non-essential for constructing a spanning tree.

Redundant edges are components of cycles within a graph. In a simple cycle, each edge might seem important for connection, but removing a redundant edge doesn’t disconnect the graph. This property is what differentiates redundant edges from crucial ones, like cut edges. They don’t form part of any spanning tree because such trees can't contain cycles — they must remain acyclic to be valid.

This understanding of redundant edges assists in simplifying graph structures and identifying which edges can be removed without affecting overall connectivity, offering a way to streamline communication networks and improve efficiency.