Combinatorics Through Guided Discovery, 2004a

48 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
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Problem 119.
(a) How do the number of spanning trees of G not containing the edge
e and the number of spanning trees of G containing e relate to the
number of spanning trees of G − e and G/e? (h)
(b) Use (G) to stand for the number of spanning trees of G (so that, for
example, (G/e) stands for the number of spanning trees of G/e).
Find an expression for (G) in terms of (G/e) and (G − e). This
expression is called the deletion-contraction recurrence.
(c) Use the recurrence of the previous part to compute the number of
spanning trees of the graph in Figure 2.3.6.
4
3
5
1 2
Figure 2.3.6: A graph.
2.3.7 Shortest paths in graphs
Suppose that a company has a main office in one city and regional offices in other
cities. Most of the communication in the company is between the main office and
the regional offices, so the company wants to find a spanning tree that minimizes
not the total cost of all the edges, but rather the cost of communication between the
main office and each of the regional offices. It is not clear that such a spanning tree
even exists. This problem is a special case of the following. We have a connected
graph with nonnegative numbers assigned to its edges. (In this situation these
numbers are often called weights.) The (weighted) length of a path in the graph
is the sum of the weights of its edges. The distance between two vertices is the
least (weighted) length of any path between the two vertices. Given a vertex v, we
would like to know the distance between v and each other vertex, and we would
like to know if there is a spanning tree in G such that the length of the path in the
spanning tree from v to each vertex x is the distance from v to x in G.
Problem 120. Show that the following algorithm (known as Dijkstra’s algorithm)
applied to a weighted graph whose vertices are labelled 1 to n gives,
for each i, the distance from vertex 1 to i as d(i).
1. Let d(1) = 0. Let d(i) =∞ for all other i. Let v(1)=1. Let v(j) =0for
all other j. For each i and j, let w(i, j) be the minimum weight of an