Graph Theory

1
Graph Theory
The objectives of this topic are to introduce to
students graph theory, and discuss its
applications to communication network design
problems. After studying this topic, the students
should

• know the basic terms and definitions of graph
  theory
• know some important graph theory theorems,
  results, and some algorithms about trees,
  shortest path problems, and flow problems
• be able to apply graph theory to the design of
  communication networks.

2
Undirected graph

• The following is an undirected graph

An example of a graph
3
Undirected graph (cont.)

• A graph is defined in terms of vertices (single
  vertex) and edges. Two vertices are connected to
  each other by an edge.
• A graph is G(X,E) where G is a graph, X is the
  set of vertices, E is the set of edges connecting
  the vertices, hence each of element of E is an
  unordered pair of two elements from X.

4
Undirected graph (cont.)

• referring to the previous example, G(X,E) where
  X1,2,3,4,5 and E (1,3), (1,4), (2,3),
  (3,5)1, (3,5)2 , note that in this example (1,3)
  is same as (3,1) for an undirected graph, to
  denote different edges d and f, subscripts are
  used as for (3,5)1, (3,5)2
• Any situation (system) that contains a set of
  elements (the vertices) and the relationship
  between pairs of elements (the edges) can be
  described by an undirected graph.

5
Parallel edges and simple graph

• Two edges connecting the same vertices are called
  parallel edges. A graph without parallel edges is
  called a simple graph.

Cardinality

• If there are m vertices and n edges, this is
  denoted by Xm and En, where S is the
  cardinality of a set S, which denotes the number
  of elements in it.

6
Complete graph

• If for every pair of vertices in a simple graph,
  there is an edge connecting them, the graph is a
  complete graph and the number of edges is
• In general for any simple graph, we have

7
Isomorphism

• A graph may be drawn in different ways by moving
  the vertices and stretching the edges, but these
  different drawings refer to the same graph.
• A graph G? (X?, E?) is isomorphic to a graph
  G(X,E) if X?X, E?E and it has identical
  relationship between the vertices and edges to G.

8
Isomorphism (cont.)

• An example

b
4
1
e
a
d
c
2
3
5
f
b
1
4
e
a
d
c
2
3
5
f
9
Directed graph

• In situations where the sense of direction is
  important when describing the relationship
  between two elements, a directed graph (digraph)
  may be used, the definition for a digraph is
  similar to that for an undirected graph, but each
  edge is represented as an ordered pair (an arc,
  an edge with an arrow) , as shown in the
  example.

an arc
a node
a
1
2
b
e
d
c
f
3
4
An example of a directed graph
10
Directed graph (cont.)

• A digraph is denoted as G(X,A), where X is the
  set of nodes, and A is the set of arcs each arc
  is an ordered pair of the nodes.
• In the above example, X1,2,3,4 and A (1,2),
  (2,1), (2,3), (2,4), (3,1), (4,3) .
• Unless specified otherwise, the following terms
  and definitions apply to both undirected and
  directed graphs.

11
Subgraph

• A graph g is a subgraph of G if all its vertices
  and edges are in G.

An example
Subgraphs of G
12
Network

• A network is a graph where each edge (or arc) is
  associated with a number (value). An example is
  as follows

-3

• The actual meanings of the numbers depend on the
  application. In general they may be positive or
  negative.
• A network is also called a weighted graph.

13
Loop

• A loop is an edge with identical end points, as
  shown

a loop

1
2
3
14
Planar graph

• A planar graph is a graph which can be drawn with
  no two edges crossing each other. An example of a
  planar graph is as follows

15
A non-planar graph example
1
2
5
3
4
an example of a non-planar graph
16
Geometric dual graph

• For any planar graph G, there exists a geometric
  dual graph G, which is constructed by placing a
  vertex in every region (separated by un-crossed
  edges) of G and connecting the new vertices by
  drawing an edge across every original edge. An
  example is as follows

1
region 1
1
2
1
2
region 2
2
3
4
3
4
The dual of a dual is always the original graph.
17
Incidence and degree

• An edge e is said to be incident to a vertex x if
  x is an endpoint of e.
• Two edges are said to be adjacent if they are
  incident to the same vertex.
• Two vertices are said to be adjacent (neighbors)
  to one another if there is an edge joining them.
  
• The degree of a vertex is the number of edges
  incident to it. In the case of a directed graph,
  the in-degree of a node is the number of arcs
  directed toward it, and the out-degree of a node
  is the number of arcs directed from it.

18
Examples
e1
e2
e3
1
2
3
4
graph 1
e4
e1
e3
e5
1
2
3
4
graph 2
e4
e2
e6
5

• In graph 1 e1 is incident to vertex 1 and vertex
  2 e2 and e3 are adjacent vertex 2 and vertex 3
  are adjacent to one another the degree of vertex
  2 is 3.
• In graph 2 the in-degree of node 3 is 2 the
  out-degree of node 2 is 1.

19
Path

• Consider any sequence x1, x2, , xk, xk1 of
  vertices (nodes). A path is any sequence of edges
  (arcs) e1, e2, , ek such that the endpoints of
  ei are xi and xi1 for i1, 2, , k vertex x1 is
  called the initial vertex, vertex xk1 is the
  terminal vertex.
• The length of a path is the number of edges
  (arcs) in the path.

20
Examples
e1
e2
e3
1
2
3
4
graph 1
a path
e4
a directed path
e3
e1
e5
1
2
3
4
graph 2
a path
e4
e2
e6
5

• In graph 1 above e1,e2, e3 form a path from
  vertex 1 to vertex 4 with length equal to 3.
• In graph 2 the arcs e1, e3, e5 form a directed
  path from vertex 1 to vertex 4. The directions of
  the arcs do not matter a path may be formed, for
  example, ase1, e2, e4, e5.

21
Simple path

• A path is called simple if no more than two edges
  are incident to any node in the path.

Flash movie examplea non-simple path

• For a simple graph, there is no need to list out
  the edges, it is sufficient to list out the
  vertices x1, x2, , xk, xk1

22
Cycle

• A cycle is a path whose initial vertex and
  terminal vertex are identical, and all the edges
  are distinct.
• A cycle is said to simple if an intermediate
  vertex (not the initial vertex nor the terminal
  vertex) appears no more than once.

Flash movie examplea non-simple cycle
Flash movie examplenot a cycle
23
Connectivity

• A graph is connected if there is a path joining
  every pair of distinct vertices otherwise it is
  called disconnected.

24
Components of a graph

• The sets of nodes in a graph with paths to one
  another are (connected) components. The edges
  between these nodes are also part of the
  components.

A graph of two components
25
Bridge

• If the deletion of an edge e yields a
  disconnected subgraph, the edge e is called a
  bridge.

Tree

• A graph is called a tree if it is connected and
  it contains no cycle.
• There is a unique path between two vertices in a
  tree.

26
Spanning tree

• A spanning tree of a graph G is a subgraph of G
  that is a tree and that includes all vertices of
  G.
• Every connected graph possesses (at least) one
  spanning tree.
• If there are m vertices in a spanning tree, there
  are m-1 edges.

Example in flash movie Not a spanning tree
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Algorithm for finding spanning tree

• Given a connected graph G(N,A)
• Step 1 Let Nn and A (an empty set) where
  n is an arbitrary node/vertex in G.
• Step 2 if NN, then stop (G(N,A) is the
  spanning tree) else go to step 3.
• Step 3 Let (i,j) in A be an arc/edge with i in
  N, j in N-N. Update N and A byGo to step 2.

Example in flash movie
28
Minimum weight spanning trees (MST)

• For a graph G(V,E) with V vertices and E
  edges, and a weight wjk for each edge (j,k), a
  minimum weight spanning tree (MST for short) is a
  spanning tree with minimum sum of edge weights.
• Any subtree(i.e., a subgraph that is a tree) of
  an MST is called a fragment.

29
Prim algorithm for finding MST

• Begin with any vertex. Select the edge of least
  weight that is incident on this vertex.
• At any intermediate iteration, a subtree is
  formed. Select the edge of least weight that
  connects some vertex in the subtree to a vertex
  not in the subtree.
• If the subtree spans all vertices, stop.
  Otherwise return to step 2.

Example in flash movie
30
Prim algorithm for finding MST reformulated

• Given a connected graph G(N,A) with weight wjk
  for each edge (j,k)
• Step 1 Let Nn and A (an empty set) where
  n is an arbitrary node/vertex in G.
• Step 2 if NN, then stop (G(N,A) is the
  spanning tree) else go to step 3.
• Step 3 Let (i,j) in A be the arc/edge with i in
  N, j in N-N with minimum wij for all i and j.
  Update N and A byGo to step 2.

Example in flash movie
31
Kruskal algorithm for finding MST

• Kruskal algorithm proceeds by building up
  simultaneously several fragments that eventually
  join into an MST.
• It starts with each node/vertex being a single
  node/vertex fragment.
• It then successively combines two of the
  fragments by using the arc/edge that has the
  minimum weight (over all arcs/edges that do not
  form a cycle when added to the current set of
  fragments).

Example in flash movie