Spanning tree

Definition

A spanning tree is a subgraph of a connected graph that includes all the vertices and is acyclic, meaning it contains no cycles. Each spanning tree connects all the vertices together with the minimum number of edges, which makes it essential for applications such as network design and optimization. A graph can have multiple spanning trees, and finding one can be done using various algorithms.

5 Must Know Facts For Your Next Test

1. A spanning tree for a graph with n vertices always has exactly n-1 edges.
2. If the original graph is disconnected, a spanning tree cannot be formed since all vertices cannot be included.
3. The process of finding a spanning tree can be implemented using algorithms like Kruskal's and Prim's, which are efficient for different scenarios.
4. Every connected graph has at least one spanning tree, and for complete graphs, the number of distinct spanning trees can be calculated using Cayley's formula.
5. Spanning trees are useful in network design as they ensure that there is a path between all nodes without any cycles, reducing redundancy.

Review Questions

• What properties must a subgraph possess to qualify as a spanning tree?
  • A subgraph must include all vertices of the original connected graph and must be acyclic, meaning it contains no cycles. Additionally, it should have exactly n-1 edges if there are n vertices. This ensures that every vertex is connected without forming any loops, maintaining efficiency in connectivity.
• Compare and contrast a spanning tree with a minimum spanning tree in terms of their characteristics and applications.
  • While both spanning trees and minimum spanning trees connect all vertices without cycles, a minimum spanning tree specifically minimizes the total edge weight in weighted graphs. Spanning trees can have varying edge weights and may not consider weights at all. The applications also differ; minimum spanning trees are often used in scenarios where cost minimization is crucial, such as in network routing, while general spanning trees may be used for simpler connectivity purposes.
• Evaluate how various algorithms for finding spanning trees differ in their approach and effectiveness based on different types of graphs.
  • Algorithms like Kruskal's and Prim's differ fundamentally in their approach to finding spanning trees. Kruskal's algorithm focuses on sorting edges by weight and adding them one by one to form the tree while avoiding cycles. Prim's algorithm starts from a single vertex and grows the tree by continually adding the smallest edge connecting the tree to an outside vertex. The effectiveness of these algorithms varies with the nature of the graph; for instance, Kruskal's algorithm is more efficient for sparse graphs while Prim's algorithm excels in dense graphs due to its reliance on adjacency matrices.

Related terms

connected graph: A connected graph is a type of graph where there is a path between every pair of vertices, meaning there are no isolated points.

minimum spanning tree (MST): A minimum spanning tree is a special type of spanning tree that has the smallest possible total edge weight among all spanning trees in a weighted graph.

forest: A forest is a disjoint union of trees, or in other words, a collection of one or more trees where any two trees do not share vertices.