This document summarizes a lab on data structures and algorithms related to graphs. It introduces graphs and their representations, as well as graph traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). It also discusses bipartite graphs. The lab objectives are to learn about graphs, DFS, BFS, and bipartite graphs. Various exercises are provided to apply these graph concepts.

1. DATA STRUCTURES AND ALGORITHMS
LAB 7
Bianca Tesila
FILS, March 2014

2. OBJECTIVES
 Graphs
 DFS
 BFS
 Bipartite Graphs

3. GRAPHS: INTRODUCTION
 Pair of sets: G = (V, E)
 V = the set of vertices
 E = the set of edges; E ⊆ V x V

4. GRAPHS: INTRODUCTION
 Directed/ Undirected

5. GRAPHS: INTRODUCTION
 Weighted/ Unweighted

6. GRAPHS: INTRODUCTION
 Cyclic/ Acyclic

7. GRAPHS: REPRESENTATION
 Adjacency Matrix
 A V x V matrix, with AdjacencyMatrix[i][j] storing whether
there is an edge between the ith vertex and the jth vertex or
not
 Linked List of Neighbours
 One linked list per vertex, each storing directly reachable vertices
!! What are the advantages and disadvantages of each of
them?

8. GRAPHS: TRAVERSALS
Depth-First Search - Go as far as you can (if you have not visited that
node yet), otherwise, go back and try another way
 There is no source vertex
 All the vertices are traversed
DFS (vertex u) {
mark u as visited
for each vertex v directly reachable from u
if v is unvisited
DFS (v)
}
!! Initially, all the vertices are marked as unvisited.

9. GRAPHS: TRAVERSALS
!! Exercise:
 Emily wants to distribute candies to N students, one by one, with a
rule that if student A is teased by B, A can receive the candy before
B.
 Given the lists of students teased by each student, find a possible
sequence to give the candies.

10. GRAPHS: TRAVERSALS
Breadth-First Search - Instead of going as far as possible, BFS tries to
search all paths.
 Uses a start vertex for the traversal: s
 Determine the minimum number of edges (shortest path
considering that all the edges have the same weight = 1)
between the source s and all the other vertices of the graph
 Not all the vertices are traversed
 BFS makes use of a queue to store visited (but not dead) vertices,
expanding the path from the earliest visited vertices
while queue Q not empty
dequeue the first vertex u from Q
for each vertex v directly reachable from u
if v is unvisited
enqueue v to Q
mark v as visited
!! Initially, all the vertices, except for the start vertex, are marked as
unvisited and the queue contains only the start vertex .

11. GRAPHS: TRAVERSALS
!! Exercise:
Let’s consider un undirected graph, representing a social network.
Given an user, display all his friends (or information about them) having
the degree <=N (N is given). A is friend with B if there is an edge
between A and B; we say that the degree of friendship is 1. Friends of
friends have the degree of friendship 2.

12. GRAPHS: BIPARTITE GRAPHS
 A graph whose vertices can be divided into
two disjoint sets such that every edge connects a
vertex in U to one in V.

13. GRAPHS: BIPARTITE GRAPHS
!! Exercise:
Check if a graph is bipartite and if so, display the
components of those two sets U and V. The graph will
be represented by the lists of neighbors.
Check your code for the following graphs:
 G1=({ 1,2,3,4,5,6,7,8,9},{ 12, 13, 45, 56, 75, 24, 58, 79,
43, 89})
 G2=({ 1,2,3,4,5,6,7,8,9},{ 12, 13, 45, 56, 75, 24, 58, 79,
43, 89,47})

14. HOMEWORK
Finish all the lab assignments.