Graphs

Lesson 22: Basic Graph Concepts
CmSc 175 Discrete Mathematics
1. Introduction
A graph is a mathematical object that is used to model different relations between objects
and processes:
Linked list
Flowchart of a program
Structure chart of a program
Finite state automata
City map
Electric circuits
Course curriculum
Definition: A graph is a collection (nonempty set) of vertices and a set (possibly empty)
of edges
Vertices (also called nodes): can have names and properties
Edges (also called links): connect two vertices, can be labeled, can be directed
Each directed edge has a start vertex and an end vertex
Loop: an edge that connects a vertex to itself
Parallel edges: two or more distinct edges with same start and end.
Example:
Vertices:
Edges:
A,B,C,D
AB, AC, BC, CD
A
B
C
D
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Same graph given in another way:
C
A
B
D
2. Basic Concepts
The variety of graphs is due to the variety of way we can connect the vertices in a
graph
Loops and parallel edges: e4 and e5 are parallel edges, e6 is a loop
A
e1
B
e3
e6
e2
C
e4
D
e5
Simple graph: a graph without loops and without parallel edges
Adjacent edges: incident on one and the same vertex, e.g. e2 and e3, e2 and e6, etc:
A
e1
B
e2
e3
e6
C
e4
D
e5
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Isolated vertex: no edges are incident on that vertex
Adjacent vertices: connected by an edge
A path from vertex x to vertex y : a list of vertices in which successive vertices are
connected by edges
Graph1:
A
B
C
D
Some paths in Graph1 are:
A B C D
A C B A C D
A B
D C B
C B A
Simple path:
No vertex is repeated.
Length of a path: the number of edges connecting the vertices in the path
Examples:
C A B is a simple path, while CA B A is not a simple path
Cycle:
Simple path except that the first vertex is equal to the last.
Examples:
C A B C, C B A C, A B C A, A C B A, B A C B, B C A B
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3. Types of Graphs
Simple graphs: Graphs without loops and without parallel edges
Connected graph:
There is a path between each two vertices
Graph1 is a connected graph.
Examples of disconnected graphs:
Graph2
Graph3
Vertices: A,B,C,D Vertices: A,B,C,D
Edges: AB, CD Edges: AB, AC
A
B
A
B
C
D
C
D
Complete graphs: Graphs with all edges present – each vertex is connected to all other
vertices.
Example:
A
B
C
D
E
Dense graphs: relatively few of the possible edges are missing
Sparse graphs: relatively few of the possible edges are present
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Directed graphs: The edges are oriented, they have a beginning and an end.
Sometimes the edges of a directed graph are called arcs.
Examples:
Graph4:
Graph5:
A
B
A
A
B
A
C
C
D
D
Graph4 and Graph5 are different graphs
Weighted graphs – weights are assigned to each edge (e.g. road map with distances)
Networks: weighted graphs or directed weighted graphs
Examples:
4
A
6
B
C
4
E
9
D
3
4
A
6
B
7
C
4
E
9
D
3
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4. Subgraphs
Subgraph K of a graph G: a graph with the following properties:
Each vertex in K is a vertex in G.
Each edge in K is an edge in G with the same end-points.
Example:
Graph:
A
B
Subgraphs:
C
D
A
A
B
C
D
D
B
C
D
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5. Spanning tree of a graph
Tree:
A graph with no cycles
Note: in a tree, when we choose a root we impose an orientation.
We may choose any node to be the root, E.G:
a
b
c
d
e
If we choose vertex a to be the root, then the tree will look like:
a
c
e
b
If we choose c to be the root, then the tree will be:
d
c
a
e
b
d
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Note also, that the graph does not specify the order of the children of a given node.
A spanning tree of a graph: a subgraph that contains all the vertices, and no
cycles
If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph.
Example:
Spanning trees for Graph1:
A
B
A
B
C
D
C
D
A
B
C
D
A tree with N vertices has N-1 edges.
A graph with less than N-1 edges is not connected.
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6. Graph representation in programming
There are two commonly used methods for graph representation:
Adjacency matrix representation
Adjacency-lists representation
The choice depends on the type of the graph – whether it is dense or not.
1. Adjacency matrix representation.
This method is used when the graph is dense – relatively few of all possible links are
missing.
The representations consists in building a matrix with number of rows equal to the
number of columns, equal to the number of graph nodes.
For the adjacency matrix A the following is true:
A(i,j) = 1 if there node i and node j are connected with an edge.
0 otherwise.
The diagonal elements A(i,i) can be set either to 1 or to 0, whatever is more appropriate
for the particular application.
Example:
Graph G: vertices A,B,C,D,E
Edges: (A,B), (A,D), (B,C),(B,E),(C,D), (C,E)
is represented by the following adjacency matrix:
A B C D E
A 1 1 0 1 0
B 1 1 1 0 1
C 0 1 1 1 1
D 1 0 1 1 0
E 0 1 1 0 1
When implementing this representation it is convenient to assign integers to the vertices.
If the number of nodes is V, the matrix requires V 2 bits of storage and V 2
initialize it.
steps to
2. Adjacency-lists representation.
This representation is used when few of all possible edges are present – i.e. the graph is
not dense. For each node a list is created that holds all nodes, connected to it.
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For the graph from the above example the adjacency list representation will be the
following:
Graph G: vertices A,B,C,D,E
Edges: (A,B), (A,D), (B,C),(B,E),(C,D), (C,E)
A: B, D
B: A, C, E
C: B, D, E
D: A, C
E: B, C
The adjacency list representation is better for sparse graphs because the space required is
O(V+E), as contrasted to O(V 2 ) for the matrix representation.
7. List of terms
Graph: A collection (nonempty set) V of vertices and E of edges. E may be empty.
Notation G(V, E).
Subgraph: Let G (V, E) is a graph. Gs (Vs, Es) is a subgraph of G iff Vs is a subset of
V, Es is a subset of E, containing those edges in E that connect the vertices of Vs in the
graph G
Loop: An edge that connects a vertex with itself
Parallel edges: distinct edges with same end-points
An edge is incident on each of its end-points
Adjacent edges: incident on one and the same vertex
Adjacent vertices: connected by an edge
Isolated vertex: no edges are incident on that vertex
Path from vertex x to vertex y : a list of vertices starting with x and ending in y in which
successive vertices are connected by edges
Simple path: No vertex is repeated.
Length of a path: the number of edges connecting the vertices in the path
Cycle: A path with no repeated vertices except for the first vertex and the last vertex,
which are the same
Tree unrooted: A graph with no cycles
A spanning tree of a graph: A subgraph that contains all the vertices, and no cycles
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Types of Graphs
Simple graphs: Graphs without loops and without parallel edges
Complete graphs: Graphs with all edges present – each vertex is connected to all other
vertices.
Weighted graphs – weights are assigned to each edge (e.g. road map with distances)
Directed graphs: The edges are oriented, they have a beginning and an end.
Sometimes the edges of a directed graph are called arcs.
Connected graph:
Undirected:
Directed:
There is a path between every two vertices.
There is a path between every two vertices in the underlying undirected graph
Disconnected graph: A graph that is not connected
A graph with N vertices and less than N-1 edges is disconnected.
DAGs – Directed Acyclic Graphs, directed graphs without cycles
Networks: Directed or undirected weighted graphs
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