Data Structures and Algorithms with Object-Oriented Design Patterns in C++

Minimum-Cost Spanning Trees

In this section we consider undirected graphs and their subgraphs. A subgraph  of a graph is any graph such that and . In particular, we consider connected undirected graphs and their minimal subgraphs  . The minimal subgraph of a connected graph is called a spanning tree:

Definition (Spanning Tree)  Consider a connected, undirected graph . A spanning tree  of G is a subgraph of G, say , with the following properties:
1. .
2. T is connected.
3. T is acyclic.

Figure  shows an undirected graph, , together with three of its spanning trees. A spanning tree is called a tree because every acyclic undirected graph can be viewed as a general, unordered tree. Because the edges are undirected, any vertex may be chosen to serve as the root of the tree. For example, the spanning tree of given in Figure  (c) can be viewed as the general, unordered tree

Figure: An Undirected Graph and Three Spanning Trees

According to Definition , a spanning tree is connected. Therefore, as long as the tree contains more than one vertex, there can be no vertex with degree zero. Furthermore, the following theorem guarantees that there is always at least one vertex with degree one:

Theorem  Consider a connected, undirected graph , where . Let be a spanning tree of G. The spanning tree T contains at least one vertex of degree one.

extbfProof (By contradiction). Assume that there is no vertex in T of degree one. I.e., all the vertices in T have degree two or greater. Then by following edges into and out of vertices we can construct a path that is cyclic. But a spanning tree is acyclic--a contradiction. Therefore, a spanning tree always contains at least one vertex of degree one.

According to Definition , the edge set of a spanning tree is a subset of the edges in the spanned graph. How many edges must a spanning tree have? The following theorem answers the question:

Theorem  Consider a connected, undirected graph . Let be a spanning tree of G. The number of edges in the spanning tree is given by

extbfProof (By induction). We can prove Theorem  by induction on , the number of vertices in the graph.

Base Case Consider a graph that contains only one node, i.e., . Clearly, the spanning tree for such a graph contains no edges. Since , the theorem is valid.

Inductive Hypothesis Assume that the number of edges in a spanning tree for a graph with has been shown to be for .

Consider a graph with k+1 vertices and its spanning tree . According to Theorem , contains at least one vertex of degree one. Let be one such vertex and be the one edge emanating from v in .

Let be the graph of k nodes obtained by removing v and its emanating edge from the graph . I.e., .

Since is connected, so too is . Similarly, since is acyclic, so too is . Therefore is a spanning tree with k vertices. By the inductive hypothesis has k-1 edges. Thus, as k edges.

Therefore, by induction on k, the spanning tree for a graph with vertices contains edges.