Question

Let \(T_{1}\) and \(T_{2}\) be spanning trees of a connected graph \(G .\) (i) If \(e\) is any edge of \(T_{1}\), show that there exists an edge \(f\) of \(T_{2}\) such that the graph \(\left(T_{1}-\\{e\\}\right) \cup\\{f\\}\) (obtained from \(T_{1}\) on replacing \(e\) by \(\left.f\right)\) is also a spanning tree. (ii) Deduce that \(T_{1}\) can be 'transformed' into \(T_{2}\) by replacing the edges of \(T_{1}\) one at a time by edges of \(T_{2}\) in such a way that a spanning tree is obtained at each stage. (This result will be needed in Chapter 7.)

Short answer

We have shown that for any edge \(e\) of \(T_1\), there is an edge \(f\in T_2\) such that replacing \(e\) with \(f\) in \(T_1\) results in a spanning tree. By iteratively applying this concept, \(T_1\) can be transformed into \(T_2\).

Step-by-step solution

Discussion on Spanning Trees and Edges

Remember that a spanning tree of a graph is a tree that includes every vertex of the graph. Therefore, for a connected graph \(G\), \(T_1\) and \(T_2\) will include every vertex of \(G\). Let \(e\) be an edge of the spanning tree \(T_1\). When we remove \(e\) from \(T_1\), it separates \(T_1\) into two disjoint subtrees, say \(S_1\) and \(S_2\).

Finding the Replacement Edge in \(T_2\)

Since \(T_2\) is another spanning tree of the same graph \(G\), it must contain some paths that are different from \(T_1\). This will include paths that connect the vertices in \(S_1\) and \(S_2\). Let's call this connecting edge \(f\).

Validation of the New Spanning Tree

We then form a new graph by adding \(f\) to \(T_1-e\). This graph is still connected and spans all vertices of the original \(G\), hence also a spanning tree. So for every edge \(e\) in \(T_1\), there is a corresponding edge \(f\) in \(T_2\) which can replace \(e\) in \(T_1\) to form a Spanning Tree.

Transformation of \(T_1\) into \(T_2\)

Using the result obtained from step 3, we can iteratively perform this replacement process. Each edge from \(T_1\) is replaced by an edge from \(T_2\) while ensuring that a valid spanning tree is maintained at each stage, effectively transforming \(T_1\) into \(T_2\) while remaining within the confines of the original graph \(G\).

Key concepts

Connected Graph

Imagine a puzzle where each piece represents a 'vertex', and the connecting parts are the 'edges'. A connected graph in graph theory is analogous to a puzzle where all pieces are connected together directly or indirectly. This means there is a path from any vertex to any other vertex in the graph. In real-world terms, think of a road map where every city (vertex) is reachable through some route (edge). Connected graphs are important for network design, transportation planning, and even social networking analysis.

The problem at hand involves proving that one puzzle configuration, or spanning tree, can be rearranged piece by piece into another, while always remaining a single connected whole. The connected nature of the graph ensures that when we interchange edges during our spanning tree transformation, we never create isolated 'islands' of disconnected vertices.

Graph Theory

Graph theory is a field of mathematics like the study of networks composed of vertices (nodes) and edges (links). Think of it as the science behind the structure and dynamics of connections - from friends within social networks to interconnected web pages on the internet. It sees many applications including computer science, biology, and sociology. What's fascinating about graph theory is its ability to model and solve complex real-life problems.

In our exercise, graph theory principles are applied to examine spanning trees within a connected graph. We use these principles to understand how structures within the graph can be modified without losing certain fundamental properties, such as connectivity, by transforming one spanning tree into another.

Spanning Trees

Let's simplify a complicated network. A spanning tree is like a minimal 'roadmap' that connects all 'cities' (in our case, vertices of a graph) together without any closed circuits or 'loops'. Technically, it's a subgraph that includes all the vertices, but only enough edges to keep the graph connected. This is crucial because in applications like constructing a minimal-cost network, you want to connect all points with the least amount of 'cable' (edges).

In the context of the exercise, we're concerned with how we can travel from one 'roadmap' (spanning tree) to another by constructing and deconstructing roads (edges) one by one. Each step must also be a complete 'roadmap' covering all cities, without creating any shortcuts or overpasses (keeping the graph loop-free and connected).

Edges and Vertices

When you're building a model, 'vertices' are like the pins holding the pieces together, while 'edges' are the rods or strings tying the pins to one another. Edges are the relationships or connections, while vertices are the entities or points. In a graph, the number of edges and vertices, as well as how they're arranged, can tell you a lot about the structure — whether it's strong, frail, complicated, or simple.

In graph transformation exercises like the one provided, acknowledging the roles of edges and vertices becomes fundamental. You're essentially keeping the 'pins' (vertices) constant while retying the 'strings' (edges) to morph one shape (spanning tree) into another, carefully maintaining the integrity of the overall model.