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6 Graphs
6.1 Objectives
At the end of the lesson, the student should be able to:
• Explain the basic concepts and definitions on graphs • Discuss the methods of graph reprentation: adjacency matrix and adjacency
list
• Traverse graphs using the algorithms depth-first search and breadth-first search
• Get the minimum cost spanning tree for undirected graphs using Prims algorithm and Kruskals algorithms
• Solve the single-source shortest path problem using Dijkstras algorithm • Solve the all-pairs shortest path problem using Floyds algorithm
6.2 Introduction
This lesson covers the nomenclature for the ADT graph. It also discusses different ways to represent a graph. The two graph traversal algorithms are also covered, as well as the
minimum cost spanning tree problem and shortest path problems.
6.3 Definitions and Related Concepts
A graph, G = V, E , consists of a finite, non-empty set of vertices or nodes, V, and a set of edges, E. The following are examples of graph:
Figure 1.77 Examples of Graph
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An undirected graph is a graph in which the pair of vertices representing an edge is unordered, i.e., i,j and j,i represent the same edge.
Figure 1.78 Undirected Graph
Two vertices i and j are adjacent if i, j is an edge in E. The edge i, j is said to be incident on vertices i and j.
Figure 1.79 Edge i, j
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A complete undirected graph is a graph in which an edge connects every pair of vertices. If a complete undirected graph has n vertices, there are nn -12 edges in it.
Figure 1.80 Complete Undirected Graph
A directed graph or digraph is a graph in which each edge is represented by an ordered pair i, j, where i is the head and j is the tail of the edge. The edges i,j and
j,i are two distinct edges.
Figure 1.81 Directed Graph
A complete directed graph is a graph in which every pair of vertices i and j are connected by two edges i,j and j,i. There are nn-1 edges in it.
Figure 1.82 Complete Directed Graph
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A subgraph of an undirected directed graph G = V,E is an undirected directed
graph G’ = V’,E’ such that V
⊆
V and E’
⊆
E.
Figure 1.83 Examples of Subgraph Figure 1.84
A path from vertex u to vertex w in an undirected [directed] graph G = V,E is a sequence of vertices v
o
, v
1
, v
2
, ..., v
m-1
, v
m
where v
o
≡ u and v
m
≡ w, such that
v ,v
1
,v
1
,v
2
,..., v
m-1
, v
m
[ v ,v
1
, v
1
,v
2
, ..., v
m-1
, v
m
] are edges in E.
The length of a path refers to the number of edges in it. A simple path is a path in which all vertices are distinct, except possibly the first and
the last.
A simple path is a simple cycle if it has the same start and end vertex. Two vertices i and j are connected if there is a path from vertex i to vertex j. If for
every pair of distinct vertices i and j there is a directed path to and from both vertices, it is said to be strongly connected. A maximal connected subgraph of an undirected
graph is known as connected component in an undirected graph. In a directed graph G, strongly connected component refers to the maximal strongly connected
component in G.
A weighted graph is a a graph with weights or costs assigned to its edges.
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Figure 1.85 Weighted Graph Figure 1.86
A spanning tree is a subgraph that connects all the vertices of a graph. The cost of a spanning tree, if it is weighted, is the sum of the weights of the branches edges in the
spanning tree. A spanning tree that has the minimum cost is referred to as minimum cost spanning tree. This is not necessarily unique for a given graph.
Figure 1.87 Spanning Tree
6.4 Graph Representations
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