Lecture 2 Graph Theory

8/13/2019 Lecture 2 Graph Theory

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Dynamics of Complex Networks and SystemsSYSM 6302 Spring 2014

Mark W. Spong

Lars Magnus Ericsson Chair and DeanThe Erik Jonsson School of Engineering and Computer ScienceThe University of Texas at Dallas

800 W. Campbell Rd.Richardson, TX 75080

[email protected]

Mark W. Spong Dynamics of Complex Networks and Systems

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Graph TheoryA Little History

The history of complex networks has its origins with Leonard Euler’sfamous solution to the Seven Bridges of Königsberg Problem.

River Bank

Island

Island

River BankCan you start at one of the river banks or one of the islands and crosseach bridge once and only once?


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Graph TheorySeven Bridges of Koenigsberg

Euler’s solution starts by representing each land mass as a single point(node) and drawing links between these nodes - each link representing a

path over a bridge connecting the two land masses.A

B

C

D

We now call this object a Graph. The number of links at each node iscalled it’s Degree. Euler proved that a solution is possible if and only if one of the following conditions is true:

The degree of each node is even, or

Exactly two of the nodes have odd degreeMark W. Spong Dynamics of Complex Networks and Systems

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Graph TheorySeven Bridges of Königsberg

Let’s discuss why this is true . . .

Thus we can see that there is no solution to the Seven Bridges of Königsberg Problem.

A

B

C

D

since all four nodes have odd degree, i.e. an odd number of links.


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Graph TheorySeven Bridges of Königsberg

As a side note, according to Wikipedia, two of the seven original bridgeswere destroyed during the bombing of Königsberg in World War II. Two

others were later demolished and replaced by a modern highway. Thethree other bridges remain, although only two of them are from Euler’stime (one was rebuilt in 1935). After World War II, the city was renamedKaliningrad. Thus, as of 2000, there are now five bridges in Kaliningrad.

It turns out that exactly two of the nodes have degree 3. The other twohave degree 2. Thus the problem now has a solution provided one beginson one island and ends on the other island.

River Bank

Island

Island

River BankMark W. Spong Dynamics of Complex Networks and Systems

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Graph TheoryBasic Concepts and Definitions

A mathematical definition of a graph can be stated as follows,

Definition (Graph)

A Graph G (more specifically, an undirected graph) is a finite setV , together with a symmetric relation E on V .

Note: by a Relation E on V is meant a subset of the Cartesianproduct V × V .

The set V = {v1, v2, . . . , vn} is called the Vertex Set of G.

The set E = {{(v1, v2)}, . . . {(vi, vj)} . . . } is called the Edge Set.

RemarkFor simplicity, we write the edge {(vi, vj), (vj , vi)} as vi, vj.We can also label the edges as E = {e1, . . . , em}


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Example

Consider the graph shown below. The Vertex Set is {v1, v2, v3, v4}.The Edge Set is {e1, e2, e3, e4, e5} = {v1, v2, v2, v3, v1, v3, v3, v4, v2, v4}

v1 v2

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v4

e1

e3 e2

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e4


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A graph may have multiple edges connecting the same vertices and may

also have Loops, which are edges connecting a vertex to itself. Such agraph is often called a Multigraph.

v1 v2

v3 v4

Illustrating a Multigraph

A graph without multiple edges between the same pair of vertices andwithout any loops is called a Simple Graph.


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Definition

The number of vertices in a graph G, denoted |V |, is called the

Order of G.

The number of edges in a graph G, denoted |E |, is called the Sizeof G.

For example, in the graph below, the Order |V | = 4 and the Size

|E | = 5.

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v4

e1

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e4


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Definition

Two vertices are said to be Adjacent if vi, vj ∈ E (G). Otherwise

vi, vj are Nonadjacent.If e = vi, vj ∈ E (G) then vi and vj are Incident to or with theedge e.

If u, v and u, w are distinct edges of G, then u, v and u, ware called Adjacent Edges.

For example, in the graph below, v1 and v2 are adjacent vertices, whilev1 and v4 are nonadjacent. Also, v1, v2 and v1, v3 are adjacent edges,whereas the edges v1, v2 and v3, v4 are nonadjacent.

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Definition (Digraph)

A Directed Graph, or Digraph, is a finite set V together with a

relation E on V , not necessarily symmetric.

Each ordered pair (u, v) ∈ E (G) is called a Directed Edge or Arc.Digraphs are drawn with arrows to indicate the directed edges.

Example

Consider the directed graph below. The arrows indicate that the edges v1, v2 and v2, v3 belong to the edge set E (G) but v2, v1 and v3, v2 are not in E (G).

v1 v2

v3


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Example

Digraphs are used when information flow or relationships are one-way, i.e.non-symmetric.

Parent

Child Child

Grandchild


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ExampleA graph is also a digraph. The diagrams below are all equivalent ways to represent the same information. Graph (a) is drawn as an undirected graph, graph (b) is drawn as a multigraph, and graph (c) is drawn as adirected graph.

v1

v2

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v4

(a)

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(b)

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Example

A city has both two-way and one-way streets. The traffic pattern may be represented as a digraph where intersections are vertices and an arc u, vmeans that it is possible to drive legally from u to v.

v1 v2 v3 v4

v5v6v7


Graph Theory

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Graph TheoryWeighted Graphs and Networks

In many applications it is useful to assign values or weights to the edgesof a graph.

Definition (Weighted Graph)

A Weighted Graph is a graph together with a function f : E → . The

function f assigns a real number (a weight) to each edge of the graph.

Remark

A weighted graph is also called a Network. Since, given any graph, we can trivially assign a weight of unity to each edge, we use the terms graph and network interchangeably.


Graph Theory

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Example

Cities v1, . . . , v4 are connected by multiple highways. The weights represent average travel times in hours.

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Graph TheoryFurther Definitions

Definition

By a ( p, g)-graph we mean a graph of order p and size g. Let v ∈ G be avertex if G. The number of edges incident with v is called the Degree of v in G, denoted deg v.

Example

v1 v2

v3 v4

v5

The degrees of the vertices v1, v2, v3, v4, v5 are, respectively 1, 2, 3, 2, 0.


Graph Theory

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Graph TheoryFurther Properties

In an undirected graph G each edge contributes to the degree sum of thetwo vertices to which it is connected. Hence the following theorem isimmediate:

Theorem

For an undirected graph G the sum of the degrees of the vertices is twice the number of edges.

Example

The graph in the previous example is seen to have four edges, and the

sum of the degrees of the vertices is 1 + 2 + 3 + 2 + 0 = 8.


Graph Theory



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As a Corollary to the previous theorem, we can state

Corollary

Every undirected graph contains an even number of odd vertices.

Proof: Let ne be the total number of edges in a graph and let de and dobe the sum of the degrees of the even and odd vertices, respectively.From the previous theorem we have do = 2 ∗ ne − de. Thus, do is aneven number. Since do is calculated as a sum of odd numbers (thedegrees of odd vertices), it must be the sum of an even number of terms.

CorollaryThe number of people on this planet who have shaken hands withan odd number of people is even.


Complex Networks

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Complex NetworksIn-Degree, Out-Degree

The previous results on vertex degree hold only for undirected graphs, in

general. For a Directed Graph we have the following:

Definition

Let G be a directed graph. The In-Degree of a vertex v ∈ G is thenumber of edges Entering v

The Out-Degree of a vertex v ∈ G is the number of edges Leaving v.The Degree of v is then the sum In-Degree + Out-Degree.

v

For example, the In-Degree of the above node v is one and theOut-Degree is two.


Graph Theory

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Definition

If every vertex of a graph G has the same degree r, we say that thegraph is r-Regular.

The graph G below is a 3-regular graph.

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DefinitionA graph is Complete if every two vertices are adjacent.

The complete graph with p vertices is typically denoted as K p. Thefigure below shows the first five complete graphs.

v1

K 1

v1

v2

K 2

v1

v2 v3

K 3

v1v2

v3 v4

K 4

v1

v2

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v4 v4K 5


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The complete graph K 6 is shown below:

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v5 v6

Theorem

A complete graph of order p is ( p − 1)-regular.The complete graph K n has

n(n−1)2 edges


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p yThe Hakimi-Havel Theorem

The Degree Sequence of a graph is a list of the vertex degrees in

descending order.

Example

v1

v2

v3

v4

The degree sequence of the above graph is therefore [3, 3, 2, 2].


Graph Theory

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p yGraphical Sequences

An interesting question is whether a given sequence of integers indescending order corresponds to a physical graph.

Such a sequence is called Graphical.

ExampleCan you draw a graph with degree sequence [3, 2, 2, 2]? What about the degree sequence [3, 2, 1]?

The following Theorem gives a constructive procedure to determine

whether or not a give sequence is graphical.


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p yGraphical Sequences

Theorem (Hakimi-Havel)

Let ν = [v1, v2, . . . , vk] be a non-increasing sequence of k ≥ 2 integers such that no component vi of ν is greater than k − 1. Let ν̃ be the

vector obtained from ν by deleting v1 from ν and subtracting 1 fromeach of the next v1 components of ν . Let ν 1 be the non-increasing vector obtained from ν by rearranging components, if necessary.Then ν is graphical if and only if ν 1 is graphical.


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p yHakimi-Havel Theorem

We can therefore use the following algorithm to determine if a givensequence of integers is graphical.

Step 0: Set ν as the current vector with k components

Step 1: If any component of ν is greater than k − 1 go to Step 5

Step 2: If any component of ν is negative go to Step 5

Step 3: If all components of ν are equal to 0, go to Step 6

Step 4: Rearrange ν so that it becomes a non-increasing sequence of integers with v1 as first component. Delete v1 and subtract 1 from eachof the next v1 components of ν . Go to Step 1.

Step 5: ν is not graphical. Go to Step 7.

Step 6: ν is graphical. Go to Step 7.

Step 7: Stop


Graph Theory

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Hakimi-Havel Theorem

Example

Let ν = [5, 4, 4, 3, 3, 3, 2]. Then the above algorithm gives

Iteration 1: ν 1 = [3, 3, 2, 2, 2, 2]

Iteration 2: ν̃ = [2, 1, 1, 2, 2]; ν

1

= [2, 2, 2, 1, 1]Iteration 3: ν 1 = [1, 1, 1, 1]

Iteration 4: ν̃ = [0, 1, 1]; ν 1 = [1, 1, 0]

Iteration 5: ν 1 = [0, 0]

Therefore, ν is graphical.


Graph Theory

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Hakimi-Havel Theorem

The graph below has the given degree sequence ν = [5, 4, 4, 3, 3, 3, 2].

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Graph Isomorphism

How can we tell when two given graphs represent the ‘same’ structure?Intuitively, two graphs are identical if it is possible to relabel and redrawone to appear identical to the other.

Example

In the figure below, it is easy to see that graphs G1 and G2 are the same,

but that G3 is fundamentally different.

v1 v2

v3 v4

(G1)

u1 u2

u3

u4

(G2)

u1 u2

u3

u4

(G3)


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Graph Isomorphism

Definition

Let G1 and G2 be graphs. An Isomorphism from G1 to G2 is aone-to-one mapping

φ : V (G1) → V (G2)

onto V (G2) such that two vertices u1, v1 are adjacent in G1 if and onlyif vertices φ(u1), φ(v1) are adjacent in G2.

We say that G1 and G2 are Isomorphic Graphs if there exists anisomorphism between them.

If φ : V (G1) → V (G2) is an isomorphism, then the Inverse mapping φ−1

exists and is an isomorphism from G2 to G1. φ−1 is defined as follows:

φ−1(v2) = v1 ∈ G1 such that φ(v1) = v2.


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Graph Isomorphism

The isomorphism relation is an Equivalence Relation on the set of graphs. This means that isomorphism relation partitions the set of allgraphs into Equivalence Classes.

Remark

It is easy to see that if G1 and G2 are isomorphic graphs then they have the same order and same size. Also, the degrees of the vertices are the same.

However, if two graphs have the same order and size and the vertices have the same degrees, they are not necessarily isomorphic.


Graph TheoryGraph Isomorphism

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Graph Isomorphism

Example

The graphs G1 and G2 shown below each have order 6 and size 9.Moreover, the degree of each vertex is the same in each graph. Yet the two graphs are not isomorphic. (How can we prove this?)

v1

v2

v3

v4

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u1 u2

u3u4

v5

v6


Graph TheoryConnectivity

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Connectivity

Connectivity is one of the most important properties of a network.Intuitively, a graph is connected if it is possible to travel between any two

vertices along edges of the graph.Example

The graph on the left is connected. It is possible to start at any vertex of the graph and reach any other vertex by following edges of the graph.The graph on the right is clearly not connected since node v1 is not

connected by an edge to any other node in the graph.

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Graph TheoryConnectedness

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Connectedness

We next give a few definitions and terminology related to traveling on

networks, i.e. moving between nodes along incident edges.Definition

A uv walk is a sequence of vertices and edges beginning with u andending with v.

A closed walk is a walk with u = v.

A trail is a walk in which all edges are distinct.

A circuit is trail in which u = v.

A path is a trail in which all vertices are distinct.

A Cycle is a circuit that does not repeat any vertices, except the

first and the last.

We will illustrate these concepts using the 3-regular graph with 6 nodes.



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Connectedness

Example

Shown below is a v1v4 walk: {v1, v2, v2, v5, v5, v4}

Since all edges in this walk are distinct, it is also a Trail.

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Connectedness

Example

Shown below is a v1v1 closed-walk: {v1, v2, v2, v5, v5, v4, v4, v1}

Since this is also a Trail, it is also a Circuit. Since no vertices are repeated, except the first and last, it is also a Cycle.

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Example

Shown below is a Path from v3 to v6.

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There are generally many paths between given nodes. Often, we areinterested in finding the shortest path between two nodes. In this case,

the shortest path from v3 to v6 is

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The figure below summarizes the relationships among the various notionsof

Walk

Trail

Circuit

Cycle

Path



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Finally, we can state the following

Definition

Two vertices are Connected if there is a uv path connecting them.

A graph is Connected if every two vertices are connected; otherwise

it is disconnected.

This matches our intuitive notion of what it means for a graph to beconnected.


Graph TheorySubgraphs



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Definition

Let G be a graph. A graph H is a Subgraph of G if V (H ) ∈ V (G) and

E (H ) ∈ E (G).If a graph F is isomorphic to a subgraph H of G, then F is also called asubgraph of G.

Example

v1

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v5 v6

v2

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v5 v6


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DefinitionA connected subgraph H of a graph G is a Component of G if H is notcontained in any connected subgraph of G having more vertices or edgesthan H .


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Example

The graph below represented by the orange vertices and bold edges is not a component. Adding any of the vertices and edges connected to them results in a larger connected subgraph of the total graph.

Remark

A connected graph has only one component.


Graph TheoryCut-Vertices and Bridges

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It is often important to know if a connected graph remains connected if a

given vertex or edge is removed.

Example

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v4 v5

v6

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v8

We see that removing the edge v4, v5 disconnects the graph. Removing any edge other than v4, v5 leaves the graph connected.


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Definition

If e is an edge of a graph G then G − e is the subgraph of G havingthe same vertex set and all edges of G except e.

If v is a vertex of G then G − v is the subgraph of G whose vertexset consists of all vertices of G except v and all edges of G except

those edges that are incident with v.An edge e is a Bridge if G − e is disconnected.

A vertex v is a Cut-Vertex if G − v is disconnected.

Theorem

Let G be a connected graph. An edge e of G is a bridge if and only if edoes not lie on any cycles of G.


Graph TheoryBridges

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Consider again the graph below.

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v4 v5

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v8

We see that only edge v4, v5 does not lie on any cycle.

Any edge other than v4, v5 can be removed without affectingconnectivity of the graph.


Graph TheoryStrongly and Weakly Connected Digraphs

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Fora directed graph the notion of connectivity is different from that of an

undirected graph. In a digraph, two vertices vi and vj may be connectedby edges but there may not necessarily be a directed path from vi to vj .

We would like to distinguish such a situation from one where there is noedge at all connecting vi and vj . We do this by introducing the notionsof Strong and Weak connectivity.

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DefinitionA directed graph is Strongly Connected if there is a directed pathbetween any two vertices. The Strong Components are themaximal strongly connected subgraphs.

A directed graph is Weakly Connected if there is an undirected

path connecting any two vertices.

In other words, strongly connected means that one can find a pathbetween vertices respecting the direction of the arrows.

Weakly connected means that on can find a path possibly ignoring thesense of the arrows, when necessary.



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The figure below illustrates the differences between weak connectivityand strong connectivity for directed graphs.

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Strongly Connected

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Weakly Connected


Eulerian and Hamiltonian Graphs

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In this lecture we will discuss Eulerian and Hamiltonian graphs.

Eulerian and Hamiltonian graphs arise in transportation problems in

operations research, in bioinformatics, in computer programmingapplications, and other applications.

Definition

(Traversable and Eulerian Graphs)

A graph (or multigraph) G is called Traversable if there is a trailcontaining all vertices and edges of G. Such a trail is called anEulerian Trail.

A circuit in a graph containing all vertices and edges is called anEulerian Circuit. (Recall that a circuit is a trail beginning and

ending at the same vertex).In this case the graph G is called an Eulerian Graph.


Eulerian and Hamiltonian GraphsEulerian Graphs

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We have already discussed the concept of Eulerian Trails and Circuits inour introduction to the Seven Bridges of Königsberg Problem. We cansummarize this as the following

Theorem

Let G be a connected graph.

G is Eulerian if and only if the degree of every vertex is even.

G is Traversable if and only if there are exactly zero or two vertices of G with odd degree.

A method for finding an Eulerian trail, when one exists, was given by theFrench mathematician M. Fleury in 1883.


Eulerian and Hamiltonian GraphsFleury’s Algorithm

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Given a traversable graph, G, Fleury’s Algorithm is relatively simple andscales linearly in the number of edges. We can state Fleury’s Algorithmas follows:

If G has exactly two odd vertices, pick one of them. Otherwise, pickany vertex, V , to start.

From V , choose any edge, e, that is not a bridge unless there is noother edge available.

Traverse the chosen edge to the next vertex.

Remove (or otherwise label) the edge, e, forming the reduced graphG

−e.

Repeat the above procedure until all edges have been traversed.


Eulerian and Hamiltonian GraphsFleury’s Algorithm

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The key point to remember is not to traverse bridge edges if other edgesare available.

Exercise

Apply Fleury’s Algorithm to the graph G below.

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v8


Eulerian and Hamiltonian GraphsApplications

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An interesting application of Eulerian Graphs is for the problem of

reconstructing DNA sequences from fragments of DNA.Pevzner, Pavel A.; Tang, Haixu; Waterman, Michael S. (2001). ”An Eulerian

trail approach to DNA fragment assembly”. Proceedings of the National

Academy of Sciences of the United States of America 98 (17)

Other applications include

Routing of garbage trucks or postal trucks

Checking a website for broken links

Mazes and Labrynths

Designing fault tolerant computer networksAll of these applications involve navigating through a graph while ideallynavigating each edge at most once.


Eulerian and Hamiltonian GraphsHierholzer’s Algorithm

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Although Fleury’s algorithm is linear in the number of edges, one mustalso factor in the problem of determining bridge edges, which increases

the computational complexity, roughly from O(|E |

) to O

(|E |

2).

In the case of Eulerian graphs, a more efficient, O(|E |), algorithm isHierholzer’s Algorithm.

Choose any starting vertex V and follow a trail of edges that returnsto V forming a circuit. Since the degree of each vertex is even, sucha circuit will always exist because, when the trail enters anothervertex U there must be an unused edge leaving U .

If all edges belong to the circuit above, it is an Eulerian circuit. If not, choose a vertex, W , belonging to the circuit that has adjacentedges not on the circuit.

Follow a new trail from W following unused edges of the graph untilreturning to W .

Join the circuit starting from W to the existing circuit.Continue in this way until all edges are used.


Eulerian and Hamiltonian GraphsExample

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Consider the Eulerian graph below

A B

C D

E F

Since the degree of every node is even, we can choose the starting nodearbitrarily. Starting, for example, from F we can follow the circuit

FCDF . Next, if we choose D, we can follow the circuit DABD.Joining this circuit to the previous one gives F CDABDF .

Next, if we choose A, we have the circuit AECA. Finally, joining this tothe previous cycle gives the Eulerian Cycle FCDAECABDF .


Eulerian and Hamiltonian GraphsHamiltonian Graphs

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Definition

A graph G is Hamiltonian if there is a cycle containing every vertex of

G.Finding Hamiltonian paths, or even knowing when one exists is a muchharder problem and there are no easy solutions known.

Example

The graph G1 below is Hamiltonian, while G2 is not. In G1 it is easy to see that u1, u2, u5, u4, u3, u1 is a Hamiltonian cycle.



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Remark

The graph G2 is not Hamiltonian.

Proof: The proof is by contradiction. Suppose that G2 is Hamiltonian.The G2 contains a Hamiltonian cycle C containing every vertex of G2. Inparticular, C contains v2, v3 and v4. Each of these vertices has degreetwo. Therefore the cycle C must contain both edges incident with eachof these vertices. In particular, C must contain the edges v1v2, v1v3 andv1v4. However, any cycle can contain only two edges incident with avertex on the cycle. Therefore, G2 is not Hamiltonian.



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Theorem

If G is a graph of order p ≥ 3 such that deg v ≥ p/2 for every vertex G,

then G is Hamiltonian.

Proof: If G has order p = 3 and deg v ≥ 3/2 for every vertex of G thenG is a triangle. Therefore the result is true for p = 3. Suppose now that

p ≥ 4. Among all paths in G, let P be a path with the largest number of

vertices. Suppose P : u1, u2, . . . , uk is this path as shown in the figurebelow.



Since no path in G has more vertices than P every vertex adjacent to u1

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Since no path in G has more vertices than P , every vertex adjacent to u1and to uk must belong to P . Since u1 is adjacent to at least p/2vertices, it follows that P must contain at least 1 + p/2 vertices.

Now, there must be some vertex ui on P , with 2 ≤ i ≤ k, such that u1 isadjacent to ui and uk is adjacent to ui−1. If this were not the case, thenfor each vertex ui adjacent to u1, the vertex ui−1 would not be adjacentto uk. However, since there are at least p/2 vertices ui adjacent to u1,there would be at least p/2 vertices ui−1 not adjacent to uk. Thereforedeg uk ≤ ( p − 1) − p/2 < p/2, which is impossible since deg uk ≥ p/2.Hence there is a vertex ui on P such that u1ui and ukui−1 are bothedges of G.



It follows that there is a cycle C containing all the vertices of P as shown

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It follows that there is a cycle C containing all the vertices of P as shownin the figure below.

If all vertices of G belong to C , then C is a Hamiltonian cycle and G is aHamiltonian graph. Suppose there is a vertex w of G that does not

belong to C . Since C contains at least 1 + p/2 vertices, fewer than p/2vertices of G do not lie on C . Since deg w ≥ p/2, the vertex w must beadjacent to some vertex uj of C . However, the edge wuj and the cycleC together produce a path having one more vertex than P , whichcontradicts the fact that P is maximal. Therefore C must contain all thevertices of G so that G is Hamiltonian.



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Remark The condition deg v ≥ p/2 for every vertex v of a graph G issufficient for G to be Hamiltonian, it is not necessary. For example, Gmay just be a cycle, hence Hamiltonian, in which case every vertex hasdegree two independent of the number of vertices.


Eulerian and Hamiltonian GraphsSome Necessary Conditions

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The conditions below are Necessary for a graph to be Hamiltonian. Inother words, if a graph is Hamiltonian, then the conditions are true.

1 Every undirected Hamiltonian graph is connected and every directed

Hamiltonian graph is strongly connected.2 Every vertex of a Hamiltonian graph has degree ≥ 2.

3 If v is a node in an undirected Hamiltonian graph G and deg(v) = 2,both edges incident on v are part of every Hamiltonian cycle of G.


Eulerian and Hamiltonian GraphsSome Sufficient Conditions

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The conditions below are Sufficient for a graph to be Hamiltonian. Inother words, if the conditions are true, then the graph is Hamiltonian.

1 Every complete, undirected graph is Hamiltonian.

2 If G is a simple graph of order n such that deg(u) + deg(v) > n − 1

for every pair of vertices u and v, then G is Hamiltonian.3 If G is a simple graph of order n such that the degree of each vertex

is at least n/2, then G is Hamiltonian.

4 If G is a simple graph of order n ≥ 3 and if deg(u) + deg(v) ≥ n forall nodes u and v that are not adjacent, then G is Hamiltonian.


Eulerian and Hamiltonian GraphsThe Traveling Salesman Problem

The Traveling Salesman Problem deals with finding a Hamiltonian

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The Traveling Salesman Problem deals with finding a Hamiltoniancycle on a network, i.e. a weighted graph. The weights represent the cost

of traversing the edges.Example

Suppose a Salesman is to visit a number of cities and return home. Theedge weights may represent distance or time. The problem is to find theHamiltonian cycle such that minimizes the total travel distance or travel

time.


Eulerian and Hamiltonian GraphsFinding Hamiltonian Cycles

The numbers of simple Hamiltonian graphs on n nodes for n = 2 3

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The numbers of simple Hamiltonian graphs on n nodes for n = 2, 3, . . .are given by 0, 1, 3, 8, 48, 383, 6196, 177083, . . . .

Hamiltonian graphs with three and four nodes

In the theory of computational complexity, the decision version of theTSP (where, given a length L, the task is to decide whether any tour isshorter than L) belongs to the class of NP-complete problems. Thus, itis likely that the worst case running time for any algorithm for the TSPincreases exponentially with the number of cities.


Eulerian and Hamiltonian GraphsTraveling Salesman Problem

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The TSP problem was first formulated as a mathematical problem in

1930 and is one of the most intensively studied problems in optimization.It is used as a benchmark for many optimization methods. Even thoughthe problem is computationally difficult, a large number of heuristics andexact methods are known, so that some instances with tens of thousandsof cities can be solved.

The TSP has several applications, such as planning, logistics, and themanufacture of microchips. Slightly modified, it appears as asub-problem in many areas, such as DNA sequencing. In theseapplications, the concept city represents, for example, customers,soldering points, or DNA fragments, and the concept distance represents

traveling times or cost, or a similarity measure between DNA fragments.In many applications, additional constraints such as limited resources ortime windows make the problem considerably harder.


Eulerian and Hamiltonian GraphsCircuit Board Drilling

A famous application of the TSP is the problem of drilling holes in a

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circuit board.

Suppose you want to automate the drilling of a number of holes in acircuit board. In order to minimize the time it takes to drill the holes, oneshould minimize the total distance traveled by the drill bit. The figurebelow shows a pattern of 2400 holes and an optimal schedule whose totaldistance is less than half of a brute force ‘raster scan.’


Graphs and MatricesAdjacency Matrix

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Definition

(Adjacency Matrix)

Let G be a simple graph of order p with vertices v1, v2, . . . , v p.Then the Adjacency Matrix A = A(G) = (aij) is the p × p matrix withaij = 1 if vertices vi and vj are adjacent and aij = 0 otherwise.

Example

Consider the graph G1 below. The Adjacency matrix A1 is given by v1 v2

v3

v4

G1 A1 =

0 1 1 01 0 1 01 1 0 10 0 1 0


Graphs and MatricesRemarks

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Remarks

Since the graph is undirected, the adjacency matrix is symmetric,i.e. aij = aji .

Also, since the graph is simple the diagonal elements of theadjacency matrix are zero and the off-diagonal elements are 0 or 1.

Moreover, the sum of the elements of row i is the degree of vertex i.

The sum of all the elements of the adjacency matrix is equal totwice the numbers of edges in the graph.

Clearly the adjacency matrix depends on how we label the vertices.

The same graph with different labels will generally give rise to adifferent adjacency matrix.


Graphs and MatricesExample

Example

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Example

Consider the graph G2 below. The Adjacency matrix A2 is given by

v1

v2

v3v4

G2 A2 =

0 1 0 01 0 1 10 1 0 10 1 1 0

The graphs G1 and G2 are isomorphic. The mapping

φ : (v1, v2, v3, v4) → (v4, v3, v2, v1)

defines an isomorphism between the graphs G1 and G2.

Note that we may also describe the relationship between V (G1) andV (G2) as a permutation on the index set (1, 2, 3, 4) → (4, 3, 2, 1).


Graphs and MatricesPermutation Matrices

Definition

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Definition

A Permutation Matrix is a matrix formed from the n × n identity

matrix by permuting the rows.

Example

Suppose π : {1, 2, 3, 4} → {1, 2, 3, 4} is a permutation given by

π(1, 2, 3, 4) = (2, 4, 1, 3)

Then the permutation matrix P π associated with the permutation π is

P π =

0 1 0 00 0 0 1

1 0 0 00 0 1 0

We note that permutation matrices are Orthogonal, i.e. P −1 = P T .


Graphs and MatricesPermutation Matrices

Th

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Theorem

Suppose two graphs G1 and G2 with adjacency matrices A1 and A2 are

given. Then G1 and G2 are isomorphic graphs if and only if there exists apermutation matrix P such that

P A1P T = A2

Back to the previous example, one can show that (1, 2, 3, 4) → (4, 3, 2, 1)is the permutation of vertex indices relating graphs G1 and G2. Formingthe associated permutation matrix P it is an easy calculation to showthat

0 0 0 1

0 0 1 00 1 0 01 0 0 0

0 1 1 0

1 0 1 01 1 0 10 0 1 0

0 0 0 1

0 0 1 00 1 0 01 0 0 0

=

0 1 0 0

1 0 1 10 1 0 10 1 1 0


Graphs and MatricesFurther Matrix Properties

An interesting property of the adjacency matrix is the following:

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Theorem

Let A be the adjacency matrix of a graph G withV (G) = {v1, v2, . . . , v p). Then the (i, j) entry of A

n gives the number of different vivj walks of length n in G.

Example

Consider again the graph G1 and adjacency matrix below v1 v2

v3

v4

G1 A1

=

0 1 1 01 0 1 0

1 1 0 10 0 1 0


Graphs and MatricesFurther Matrix Properties

3



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Then a simple calculation shows that A31 is given by

A31 =

2 3 4 13 2 4 14 4 2 31 1 3 0

Thus, for example, there are three walks of length three between thevertices v1 and v2.

v1 v2

v3

v4

G1


Graphs and MatricesAdjacency Matrix of a Digraph



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If the graph G is directed, i.e. a digraph we define the (i, j) entry of theadjacency matrix to be 1 if and only if there is an arc from vertex i tovertex j.

In this case the adjacency matrix is not necessarily symmetric.

The sum of the entries of any row of the adjacency matrix is the

outdegree of the vertex. The sum of the entries of any column is theindegree of the corresponding vertex.

Definition

Let G be a digraph. The indegree of a vertex v is the number of arcs

entering v. The outdegree of v is the number of arcs leaving v.


Graphs and MatricesIncidence Matrix

Another useful matrix characterization of a graph is the so-called



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Another useful matrix characterization of a graph is the so calledIncidence Matrix.

Definition

The Incidence Matrix of a graph G of order n and size m is the n × mmatrix M = (mi,j) with mi,j = 1 if vertex i is incident with edge j andzero otherwise.

Example

The figure below shows a graph and corresponding incidence matrix.

v1 v2

v3v4

e1

e

2

e3

e

4

e5

M =

1 0 0 1 01 1 0 0 10 1 1 0 00 0 1 1 1


Graphs and MatricesGraph Laplacian

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Next we define the Laplacian or Laplacian Matrix of a graph as follows:

Definition

Given a simple graph with n vertices, the Laplacian Matrix, L = (ij) isdefined as

ij =

deg(vi) if i = j

−1 if i = j and vi is adjacent to vj0 otherwise

It is easy to show that the Laplacian L = D − A, where D is a diagonalmatrix with the vertex degrees on the diagonal and A is the adjacency

matrix. The Laplacian is useful in several ways; for determining thenumber of connected components of the graph, determining the numberof spanning trees and other applications.


Graphs and MatricesExample

Example

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Consider the graph and associated Laplacian matrix below:

L =

2 −1 0 0 −1 0−1 3 −1 0 −1 0

0 −1 3 −1 −1 00 0 −1 3 0 −1

−1 −1 −1 −1 4 0

0 0 0 −1 0 −1

Remark

It is easy to see that the sum of the elements in each row is equal to

zero.This means that the vector x = (1, 1, . . . , 1)T satisfies Lx = 0.

As a consequence, the Laplacian is never invertible and 0 is alwaysan eigenvalue of the Laplacian.


Graphs and MatricesEigenvalues of the Adjacency and Laplacian Matrices



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Both the Adjacency and the Laplacian matrices are symmetric.

This means that all of their eigenvalues are real.

Definition

The Spectral Radius ρ(G) of a graph G is the largest eigenvalue of the

adjacency matrix A(G).The Spectral Gap σ(G) of a graph G is the largest eigenvalue of theLaplacian matrix L(G).

Later we will use the spectral radius to explain, for example, howepidemics travel through a network and we will use the spectral gap toexplain stability, or lack of it, in a network.


Graphs and MatricesAdditional Properties of the Laplacian

A direct calculation shows that

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A direct calculation shows that

xT

Lx =

xixj∈E (G)(xi − xj)

2

≥ 0

As a consequence, the Laplacian matrix is Positive Semi-definite.

This means that the eigenvalues of L are non-negative.

Let 0 = λ0 ≤ λ1 ≤ · · · ≤ λn be the n eigenvalues of L ordered fromsmallest to largest.

The multiplicity of the zero eigenvalue is the number of connectedcomponents of G.

If G is connected, therefore, λ0 is the only zero eigenvalue.λ1 is called the algebraic connectivity of G and is a measure of how well the overall graph is connected.


Graphs and MatricesLaplacian Matrix for a Complete Graph

For the complete graph K n with n vertices, each diagonal entry of theLaplacian matrix is n − 1 and all other entries are equal to −1. From

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p qthis we can show that the nonzero eigenvalues, λ1, . . . , λn−1 are all equal

to n, the order of the graph.Proof

The eigenvectors associated with eigenvalues λ1, . . . λn−1 form ann − 1-dimensional subspace orthogonal to the vector x = [1, 1, . . . , 1].Let v be any vector orthogonal to x. Then

ni=1

vi = 0

Computing the j-th entry of Lv gives

(Lv)j = (n − 1)vj −i=j

vi = nvj

Therefore, n is an eigenvalue and any vector orthogonal to x

is an ei envector for n Mark W. Spong Dynamics of Complex Networks and Systems

TreesIn this lecture we will discuss Trees, which form an important subclass of graphs.

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Trees find applications in efficient data storage and search methods in

computer science, in transportation networks, in project management(PERT), and a host of other applications.

Definition

A Tree is a connected simple graph with no cycles. Note: Without the

requirement of being connected, the graph is called a Forest.


TreesBasic Results

W ill i f b i th b t t h f

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We will now give a sequence of basic theorems about trees, whose proofsare fairly straightforward.

Theorem

For any connected, simple, graph G with n vertices and m edges,n ≤ m + 1.

Proof: We can equivalently show m ≥ n − 1. Clearly, m = n − 1 for agraph with n = 2 and m = 1. It is also clear that any connected graphcan be built from a graph with n = 2 and m = 1 by sequentially addingnodes and edges one at a time. To maintain connectivity, any time anode is added, one edge must be added. On the other hand, an edge can

be added between two existing vertices, thereby creating a cycle, withoutincreasing the number of vertices. Therefore, the number of edges isalways greater than or equal to n − 1.


TreesBasic Results

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From the proof of the previous theorem it is clear that m = n − 1 for atree, since there are no cycles. Likewise, one can easily show

Theorem

Any connected simple graph with m vertices and m edges for whichn = m − 1 is a tree.

A graph is a tree if and only if there is exactly one path between any pair of vertices u and v.

An edge e of a graph G is a cut edge if and only if e is not part of any cycle of G.

A connected graph G is a tree if and only if every edge is a cut edge.


TreesSpanning Trees

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Definition

Let G be a connected graph. An acyclic connected subgraph S of Gcontaining all of the vertices of G is called a Spanning Tree.

Graph G and Spanning Tree H (bold edges)


TreesMinimal Connector Problem

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The Minimal Connector Problem or Minimal Spanning Tree

Problem is to find a spanning tree T in a network N whose value (thesum of values of its edges) is minimum.

Kruskal’s Algorithm

Given a network N choose any edge e1 of N of minimal value. Next

choose edge e2 to be any remaining edge of minimal value. For e3choose any remaining edge of minimal value which does not form a cyclewith the previous edges. We continue this procedure until a spanning treeis obtained. A spanning tree arrived at in this way is called an EconomyTree.

Note that there is no requirement that an edge chosen at a given stepmust maintain connectivity.


TreesExample

Example

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Example

Consider the connected network N below.

Mark W Spong Dynamics of Complex Networks and Systems

TreesStep 1

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TreesStep 2

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TreesStep 3



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TreesStep 4

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TreesStep 5

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TreesStep 6

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TreesStep 7

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TreesThe Resulting Economy Network

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TreesSolution of the Minimal Connector Problem

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TheoremLet G be a connected network and let T be an economy tree of G. ThenT is a spanning tree whose value is minimal.

Proof: We need to show that if T 0 is any other spanning tree of G then

the value of T 0 is at least as large as the value of T . Denote by f (e) thevalue of edge e. Then the value of the tree T is

f (T ) =

f (e)

where the summation is taken over all of the edges of T .



If the network G has order p, then the economy tree T has p − 1 edges.L h d f T b d d

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Let the edges of T be ordered as e1, e2, . . . , e p−1.

If T 0 and T are not identical, let ei be the first edge of T in the abovesequence not in T 0.

Add the edge ei to T 0 obtaining a network G0. Suppose ei = uv.

Then a u − v path P exists in T 0 so P together with ei forms a cycle C

in G0.

Since T contains no cycles let e0 be an edge in C that is not in T .

The graph T

0 = G0 − e0 is also a spanning tree of G and

f (T

0) = f (T 0) + f (ei) − f (e0)



Si f (T ) ≤ f (T

) it f ll th t

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Since f (T 0) ≤ f (T 0) it follows that

f (e0) − f (ei) = f (T 0) − f (T

0) ≤ 0

i.e. f (e0) ≤ f (ei). However, given the above construction, ei is an edgeof smallest value that can be added to the edges e1, . . . , ei−1 without

producing a cycle. Also if e0 is added to the edges e1, . . . , ei−1 no cycleis produced either. Therefore f (ei) = f (e0) so that f (T 0) = f (T

0). Wehave shown the existence of a spanning tree T

0 of minimum value suchthat the number of edges common to T

0 and T exceeds the number of edges common to T 0 and T by one edge, namely ei. By continuing thisprocedure we can construct a spanning tree with minimum value identical

to T . Therefore the economy tree T has minimum value.

M k W S D i s f C le Net ks d S ste s

TreesPERT and the Critical Path Method

The management sciences contain many examples of very complexplanning and scheduling problems. Graph theory can be useful tosystematically analyze and plan large scale projects Techniques referred

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systematically analyze and plan large-scale projects. Techniques referred

to as PERT (Program Evaluation and Review Technique) and CPM(Critical Path Methods) use Activity Digraphs to model projectactivities.


TreesExample

Let’s look at a simple example. Suppose we want to convert a basement

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p p pp

area into a game room. The work involves four activities, paneling thewalls, installing a ceiling, carpeting the floor, and assembling a pooltable. To make things easier the carpet should be installed beforeassembling the pool table.

Activity T ime(hours)

A Assemble the pool table 4B Carpet the floor 6C Panel the walls 6D Install the ceiling 8

An Activity digraph is constructed from the list of activities by adding

nodes S (Start) and E (End) to the list of nodes together with theactivities A, B,C, D.


TreesActivity Digraph



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The vertex S is directed to a vertex v if activity v may start without anyother activity first being completed. The vertex w is directed to E if noactivity requires that w be completed before that activity begins. Finally,

vertex x is directed to vertex y if and only if no activity needs to intervenebetween the completion of x and the beginning of y. The weight of edgevivj corresponds to the time needed to complete activity vj .


TreesCritical Path

Definition

A longest path (in terms of time) in an activity digraph is called a

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A longest path (in terms of time) in an activity digraph is called a

Critical Path. The length of the critical path equals the minimum timenecessary to complete the project.

The above activity digraph has exactly one critical path S, B, A, E andthe time length is 10 hours. In practice, the problems are obviously much

more complex and there are software packages available to constructPERT charts.

Example

Explain why the graph below cannot be an activity digraph.


Graph TheoryBipartite Graphs

Definition

A Bipartite Graph is a special type of graph G whose vertices can be

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A Bipartite Graph is a special type of graph G whose vertices can be

partitioned into two sets V 1 and V 2 so that every edge of G joins avertex of V 1 to a vertex of V 2 and no vertex is connected to anothervertex in its own set.

Example

The graph below is an example of a bipartite graph

v1 v2 v3 v4 v5

u1 u2 u3 u4 u5



Theorem

A graph is bipartite if and only if it contains no odd cycles.

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Corollary

Since trees have no cycles we can state immediately that every tree isbipartite.

Example

The graphs below show a tree and the corresponding bipartite representation.

v1

v2

v3

v4 v5 v6 v7 v1 v4 v5 v6 v7

v2 v3


Graph TheoryGraph Embeddings

The way a graph is represented is often important as a way to gain

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insight into its structure.By a Graph Embedding we mean a particular drawing or representationof a graph.

Some standard way to represent a graph include:

circular embeddingranked embedding

spring embedding

lattice embedding


Graph TheoryGraph Embeddings

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The figure below shows several different embeddngs of a cube


Graph TheoryRanked Embeddings

Bipartite graphs are sometimes drawn as Ranked Embeddings. Considerthe circular graph shown below. Although it is not obvious, this isactually a bipartite graph We can show this by considering the distance

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actually a bipartite graph. We can show this by considering the distancebetween each vertex v and all other vertices. Here, by distance betweenvertices we mean the length of the shortest path connecting them.



Now, select an arbitrary vertex v and select all vertices at distance 1 fromv, then at distance 2 from v and so on. We draw the vertices at a givenfixed distance from v along a horizontal line as shown below. One can

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see that there are no edges between vertices at the same distance from v.Hence the graph is bipartite; one subset consisting of an even distance tov and the other consisting of vertices at an odd distance to v.


Graph TheoryExercise

Exercise

Consider the circular graph below

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1

23

4

5

6 7

8

Show that this can be redrawn as the bipartite graph below

1

2

3

4

5

6

7

8

Note that this is another embedding of the cube.


Graph TheoryApplications

Bipartite graphs are useful for modeling so-called matching problems.An example of bipartite graph is a job matching problem. Suppose wehave a set N of people and a set M of jobs with not all people suitable

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have a set N of people and a set M of jobs, with not all people suitablefor all jobs. We can model this as a bipartite graph (N , M , E ). If aperson ni is suitable for a certain job mj there is an edge between ni andmj in the graph.

Bipartite graphs are also extensively used in modern Coding theory,especially to decode codewords received from the channel.

In computer science, a Petri net is a mathematical modeling tool used inanalysis and simulations of concurrent systems. A system is modeled as abipartite directed graph with two sets of nodes: A set of place nodesthat contain resources, and a set of event nodes which generate and/or

consume resources. Petri nets utilize the properties of bipartite directedgraphs and other properties to allow mathematical proofs of the behaviorof systems while also allowing easy implementation of simulations of thesystem.


Graph TheoryPlanar Graphs

Definition

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A Planar Graph is a graph such that no two edges intersect.

Example

Note that one must be careful when determining whether or not a graphis planar. A seemingly non-planar graph can sometimes be redrawn so that it is planar.

The above two graphs are isomorphic and so both are planar.



Applications of Planar Graphs

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Applications of Planar Graphs

Planar graphs arise in Transportation Networks. If a graphcorresponding to a transportation network is planar then there is noneed for tunnels and bridges.

In the design of electrical circuits, planar networks mean that wires

do not have to cross. Complex computer chips are designed inlayers so that each layer is represented as a planar graph. There area number of other important applications of planar graphs.

We will next give some interesting characterizations and propertiesof planar graphs.


Graph TheoryEuler’s Formula

A planar graph encloses a number of Regions, r1, . . . , rn, where wecount the region exterior to the graph as region r1. Given such a planar

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graph with n vertices, m edges and r regions, the relationship amongthese quantities was formulated by Leonard Euler in his famous formula:

n − m + r = 2



Proof: The proof is by induction on r, the number of regions. If r = 1then there can be no regions enclosed by edges of G. Thus G must be

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then there can be no regions enclosed by edges of G. Thus G must beacyclic in which case m = n − 1. Thus n − m + r = n − (n − 1) + 1 = 2.Therefore Euler’s formula is true for r = 1.

Now assume that the formula is true for all planar graphs with less than rregions and choose a graph with r > 1. Choose an edge e, which is not

a cut edge, and let G1 = G − e. Since e is part of a cycle, removing eresults in two merged regions, reducing the number of regions by 1. ForG1, therefore, we know the formula is true. Therefore,

V (G1) − E (G1) + (r − 1) = 2

Since V (G1) = V (G) and E (G1) = E (G) − 1. Substituting these intothe above equation gives the result.



Euler’s formula can be used for, among other things, determiningwhether a given graph is planar.

Theorem

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For any connected simple planar graph G with n ≥ 3 vertices and medges, we have that m ≤ 3n − 6.

Proof: Consider a region f in any planar graph G. For any interiorregion, let B(f ) denote the number of edges by which f is enclosed, i.e.the length of its “border.” Clearly, B(f ) ≥ 3. However, with n ≥ 3 wealso have that the exterior region is “bounded” by at least 3 edges.Therefore, if there are a total of r regions, then

B(f ) ≥ 3r. On the

other hand, it is easy to see that

B(f ) counts every edge in G once ortwice and hence B(f ) ≤ 2m so that 3r ≤ B(f ) ≤ 2m and sor ≤ 23m. From Euler’s formula, therefore, we havem = n + r − 2 ≤ n + 23m − 2 so that m ≤ 3n − 6.

Note that the above result means that, if we have a simple graph inwhich m > 3n − 6, then it cannot be planar.



Example

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The complete graph K 5 on five vertices is non-planar.

v1

v2

v3

v4 v4

K 5

In the graph K 5, n = 5 and m = 10 so 3n − 6 = 9 > m and so the graph cannot be planar.



Example

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Is the complete graph K 4 planar? v1v2

v3 v4

K 4

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Documents

Lecture 2 Graph Theory