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Algorithm 15.4 The Recursive Depth-First Search (DFS) Algorithm (Preconditions: G = (V,E) is a graph or digraph with initial vertex x; each vertex has a boolean visited field initialized to false; T is a global set of edges; L is a global list of vertices.) (Postcondition: L lists the vertices in DFS order, and T is a DFS spanning tree for G.) 1. Mark x visited. 2. Add x to L. 3. Repeat steps 4 5 for each unvisited vertex y that is adjacent to x. 4. Add the edge xy to T. 5. Apply the DFS algorithm to the subgraph with initial vertex y.

CHAP. 15]

Table 15.3 shows a trace of Algorithm 15.4 on the same graph as in Example 15.26 (Figure 15.29 on page 301). The start vertex v0 = A. The result, of course, is the same as that in Example 15.27. The only real difference is that the explicit

4

Here is a Java implementation of the two traversal algorithms for the Network class introduced in Example 15.25 on page 299:

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public class Network { Vertex start; private class Vertex { Object object; Edge edges; Vertex nextVertex; boolean visited; } private class Edge { Vertex to; int weight; Edge nextEdge; } public static void visit(Vertex x) { System.out.println(x.object); }

public void breadthFirstSearch() { if (start == null) { return; } Vector queue = new Vector(); visit(start); start.visited = true; queue.addElement(start); while (!queue.isEmpty()) { Vertex v = queue.firstElement(); queue.removeElementAt(0); for (Edge e = v.edges; e != null; e = e.nextEdge) { Vertex w = e.to; if (!w.visited) { visit(w); w.visited = true; queue.addElement(w); } } } } public void depthFirstSearch() { if (start != null) { depthFirstSearch(start); } } public void depthFirstSearch(Vertex x) { visit(x); x.visited = true; for (Edge e = x.edges; e != null; e = e.nextEdge) { Vertex w = e.to; if (!w.visited) { depthFirstSearch(w); } } } }

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In This :

This uses the recursive version of the depth-first search. That requires the depthFirstSearch() method with zero parameters to start the recursive depthFirstSearch() method.

15.1 15.2 15.3 15.4 15.5 What is the difference between a graph and a simple graph In an undirected graph, can an edge itself be a path What is the difference between connected vertices and adjacent vertices Using only the definition of graph isomorphism, is it easier to prove that two graphs are isomorphic or to prove that two graphs are not isomorphic Why Are the five conditions in Theorem 15.4 on page 289 sufficient for two graphs to be isomorphic

CHAP. 15]

7:

15.6 15.7

Consumption Investment Net Exports Government Taxes and Expenditures True or False Questions Solved Problems Consumption

Why is it that the natural definition of a simple graph prohibits loops while the natural definition of a digraph allows them True or false: a. If a graph has n vertices and n(n 1)/2 edges, then it must be a complete graph. b. The length of a path must be less than the size of the graph. c. The length of a cycle must equal the number of distinct vertices it has. d. If the incidence matrix for a graph has n rows and n(n 1)/2 columns, then the graph must be a complete graph. e. In an incidence matrix for a digraph, the sum of the entries in each row equals the indegree for that vertex. f. The sum of all the entries in an incidence matrix for a graph is 2|E|. g. The sum of all the entries in an incidence matrix for a digraph is always 0. A graph (V, E) is called dense if |E| = (|V|2), and it is called sparse if |E| = O(|V|). a. Which of the three representations (adjacency matrix, incidence matrix, or adjacency list) would be best for a dense graph b. Which representation would be best for a sparse graph Why is it that, in the incidence matrix of a simple graph, there are always exactly two 1s in each column