Question

(i) Let \(C^{*}\) be a set of edges of a connected graph \(G\). Show that, if \(C^{*}\) has an edge in common with each spanning tree of \(G\), then \(C^{*}\) contains a cutset. (ii) Obtain a corresponding result for cycles.

Short answer

The first part of the exercise establishes that the set \(C^{*}\) must contain a cutset if it has at least one edge in common with every spanning tree in a graph. The second part of the exercise demonstrates that if a set of edges intersects every cycle in a graph, then it must contain an edge that breaks the cycle.

Step-by-step solution

Part I: Understanding Definitions

Let's start with the basics. A spanning tree of a graph G is a subgraph that includes all vertices of G and is also a tree. A cutset in a graph is a set of edges whose removal leaves the graph disconnected.

Part I: Proving \(C^{*}\) contains a cutset

Let's assume, on the contrary, that no subset of \(C^{*}\) forms a cutset of \(G\). Take any spanning tree \(T\) of \(G\). Then \(T\) has at least one edge (say \(e\)) in common with \(C^{*}\). Since \(C^{*}\) doesn't have a cutset, the graph \(T-e\) remains connected. But this contradicts the property of a tree, that deleting any edge from it would disconnect the tree. Hence, our assumption is incorrect, which means \(C^{*}\) must contain a cutset.

Part II: Understanding cycles

A cycle in a graph is a sequence of vertices where the first and last vertices are the same, there are no repeated edges, and there are no repeated vertices. A cycle represents the path where we can start from a point and return to the same point.

Part II: Obtaining a corresponding result for cycles

Similar to the first part, consider any cycle \(C\) in graph \(G\). If there’s a set of edges \(C^{*}\) which intersects every cycle, then \(C^{*}\) must have one edge say \(e\) in every cycle. If we remove \(e\), then \(C\) is no longer a cycle, which contradicts the assumption that \(C\) is a cycle. Therefore, \(C^{*}\) contains an edge which breaks the cycle.

Key concepts

Spanning Tree

In the world of graph theory, a spanning tree is a fundamental concept. Think of it as the skeleton of a connected graph. It includes all the vertices of the original graph but with the minimum number of edges needed to connect all vertices without forming any cycles.

• A spanning tree for a graph with \(n\) vertices will always have \(n-1\) edges.
• It's important because it connects all parts of the graph in the most streamlined way possible.
• In applications, spanning trees are useful for designing efficient networking, such as minimizing the length of cables needed for a computer network.

By focusing on the essence of the graph without extra loops, spanning trees let us think clearly about the essential connections in a network.

Cutset

A cutset represents a set of edges whose removal will split a connected graph into two or more separate components.

• Consider it like a barrier that disconnects parts of a network when removed.
• If a graph is split, it no longer has a single path between some vertices.
• In practice, identifying cutsets helps in determining weak links in a network, where failure could cause disconnection.

The original exercise deals with understanding how sets of edges, that appear in every spanning tree, affect the structure of the graph. If such a set exists, it inherently forms a cutset since its removal would isolate parts of the network, contradicting the concept of a spanning tree which should keep the graph connected.

Cycle

A cycle in a graph is like going on a walk that starts and ends at the same point, without retracing your steps. It’s a closed loop in the system. Cycles are essential to understanding graph redundancies.

• Removing a complete cycle edge by edge eventually breaks the loop.
• Each cycle offers a potential for repeated paths in the system.
• Cyclic structures are crucial in identifying feedback loops within networks.

In the exercise's context, cycles demonstrate how certain edges, common to all cycles, function as critical links. If a such set, termed \(C^{*}\), interacts with each cycle, removing any member disrupts the cycle, confirming that the cycles depend on these edges for closure.

Connected Graph

A connected graph is a robust concept in graph theory, ensuring that from any vertex you can reach any other vertex through a series of edges.

• Every connected graph has at least one spanning tree.
• It’s complete if there exists a path between every pair of vertices.
• Connectedness is essential for system designs where communication or travel between all points is crucial.

By examining edge sets in connected graphs, like in the original problem, one can determine points of vulnerability and strength in networks. When certain edges are guaranteed to intersect every spanning tree, their removal can cause disconnection, showing inherently that these are critical to maintaining the graph's connected nature.