Computing Fundamentals 2Lecture 1

A Theory of Graphs

Lecturer: Patrick BrowneRoom [KA] - 3-020, Lab [KA] - 1-017

Based on Chapter 19. A Logical approach to Discrete Math By David Gries and Fred B. Schneider

http://www.comp.dit.ie/pbrowne/compfund2/compfun2.htm

A Theory of Graphs

• We all have an intuitive understanding of graphs.

A Theory of GraphsQuestion: Starting at Newport is it possible to travel each road exactly once and return to Newport?

Some terminology

• In graph theory terms, the dots on the map are called vertices and the lines that connect the vertices are called edges. We have labelled the edges e1..e13. If we travel from Newport to Cork we call this a path from Newport to Cork.

• The graph can be written as: G = <V,E>• V = {Newport, Sligo, Athlone, Drogheda, Dublin,

Carlow, Waterford, Cork, Limerick, Galway}• E = {e1, e2, … e13}

• This is an undirected graph, there are no one-way streets.

Rephrasing the question

• Question: Starting at Newport is it possible to travel each road exactly once and return to Newport?

• Our problem can be restated:– Is there a path from vertex Newport to vertex

Newport that traverses each edge exactly once?

Reasoning about the question

• Suppose there was such a path consider the vertex Limerick. Each time we arrive at Limerick on some edge , we must leave using a different edge. Furthermore, every edge that touches Limerick must be used.

Reasoning about the question

• Thus the edges at Limerick occur in in/out pairs. It follows that an even number of edges must touch Limerick. Since three edges touch Limerick (e6,e9,e11) we have a contraction. Therefore, there is no path from Newport to Newport that traverses every edge once. We have proved this using graph theory.

Algorithms in proofs

• An algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will proceed through a well-defined series of successive states, possibly eventually terminating in an end-state1.

• The proof of some results in graph theory may involve an algorithm to construct some object like a path from one vertex to another. In Chapter 19 of Gries and Schneider algorithms are used in proofs2

Graphs and Multigraphs

• (19.1) Definition: Let V be a finite, non empty set and let E be a binary relation on V. Then G=<V,E> is called a directed graph or a digraph. An element of V is called a vertex; an element of E is called an edge.

• The binary relation E is defined as EV×V.• In figure 19.1 E and V are: • E = {<b,b>,<b,c>,<b,d>,<c,e>,<e,c>,<e,d>}• V = {a,b,c,d.e}

• A digraph G=<V,E> is loop free iff E is an irreflexive relation (no <b,b> ).

V = {a,b,c,d.e}

E = {<b,b>,<b,c>,<b,d>,<c,e>,<e,c>,<e,d>}

A directed graph or digraph c

a

d

b

e

d

c

b

a d

c

b

a

An undirected graph represented as a digraph

Fig. 19.1

Fig. 19.2bFig. 19.2 a

Here E is a set of unordered pairs i.e {a,b} rather than <a,b>. {a,b}, {a,c}, {b,c},

{b,d} these are sets{a,b} could be written <a,b>, <b,a>.

There is a one-to-one correspondence between undirected graphs and digraphs with symmetric relations.

Aside: Bags

• The elements of a set are distinct. Sometimes we need to allow duplicates. For example, if we wanted to count the people in Dublin using a list of their names we would have to allow for duplicate names, a set would not be appropriate. Mathematically a bag is a set with duplicates therefore many of the familiar set operations apply to bags.

Graphs and Multigraphs

• A multigraph is a pair <V,E>, where V is a set of vertices and E is a bag of undirected edges.

• In a multigraph there can be many edges between two given vertices. Every undirected graph is a multigraph (but not visa versa).

• Multigraphs do not depict relations.• In order to distinguish edges in a multigraph we

can label the edges.

Graphs and Multigraphs

Euler’s 7 bridges of Konigsberg, representing bridges as edges, land as vertices.

               →                →               

Because a multigraph consists of a bag of undirected edges we need labels to distinguish edges. This means that they may be parallel edges between two given vertices.

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45

6

7

The Degree of A Vertex and Graph

• The indegree of a vertex of a digraph is the number of edges for which it is an end vertex.

• The outdegree is the number of edges for which it is a start vertex.

• The degree of a vertex, written deg.v, is the sum of the indegree and the outdegree.

• A vertex whose degree is 0 is called isolated.• The degree of a vertex of an undirected graph or

multigraph is the number of edge ends attached to it.

The Degree of A Vertex and Graph

• Each edge contributes 1 to the degree of two vertices (which in a loop may be the same vertex, thus adds two to that vertex).

• (19.3) Theorem. The sum of the degrees of the vertices of a digraph or multigraph is 2 * #E (E is a set, see next slide).

• (19.4) Corollary. In a digraph or a multigraph the number of vertices of odd degree is even.

The Handshaking Theorem

Theorem: Let G=(V,E) be an undirected graph. Then twice the number of edges is equal to the sum of the degrees of V.

Number of edges = 7

1 + 2 + 3 + 2 + 2 + 3 + 1 = 14

In any group of people the number of people who have shaken hands with an odd number of other people from the group is even. Zero is an even number

Even degree

• Loop edge gives vertex a degree of 2.• In any graph the sum of degrees of all vertices equals

twice the number of edges.• The total degree of a graph is even.• In any graph there are even number of vertices of odd

degree.

Paths• The land masses are modelled as vertices. The bridges

are modelled as edges. Is it possible to tour Königsberg in a way that would traverse each bridge (edge) exactly once?

Deg(d) = 5Deg(a) = 3Deg(b) = 3Deg(c) = 3--------------- 14#Edges = 7(2 x #Edges) = 14

Fig. 19.3(b)

Aside: A theory of Sequences

• A sequence is a finite list of elements from some set. The theory of sequences can be used to reason about graphs, lists and arrays.

• Sequences are ordered, but not necessarily sorted. Like runners in a race there is a first, second and third runner. Their finishing position in the race does not usually depend on their race number. At the finish runners are not (usually!) sorted by race number.

Aside: A theory of Sequences

• Catenation: When we join two sequences together we want to preserve the position of each element. This can be illustrated by using the CafeOBJ sequence module.

• red [1 , 2 ] ^ [ 1, 2, 3 , 4 ] ^ [] ^ [0]

• Giving

• [1, 2, 1, 2, 3, 4, 0]• Note that duplicates are allowed.

Path

• A path of a multigraph or a digraph is a sequence of vertices and edges that would be traversed when walking the graph from one vertex to another, following the edges, but with no edge traversed more than once. In a directed graph an edge can only be traversed in one direction.

Paths : Example

• A path that starts at vertex b and ends at vertex c is called a b-c path. Here represented as a sequence of vertices and edges.

c

a

d

b

e

Fig. 19.1

• Path1: <c,<c,e>,e,<e,d>,d>

• Path2: <b,<b,c>,c,<c,e>,e,<e,c>,c> • Non-Path: <b,<b,c>,c,<c,e>,e,<e,c>,c,<c,e>,e>

Path length, alternative notation

Fig. 19.3(b)

The length of a path is the number of edges in it.

Path3 : <a,6,b,5,d,1,a> Length = 3 edges

Path intermediate summary

1. A path starts with a vertex, ends and alternates between vertices and edges.

2. Each directed edge in a path is preceded by its start vertex and followed by its end vertex. An undirected edge is preceded by one of its vertices followed by the other.

3. No edge appears more than once.

4. The length of a path is the number of edges on it. The path with just one vertex <a> has length zero. A self loop <b,b> is a cycle of length one.

Redundant Notation

• The notation (<a,6,b,5,d,1,a>) can be used for all graphs (directed, undirected and multigraphs).

• While the notation is necessary for multigraphs it is redundant for directed and undirected graphs (even though they are simple by 2 multigraphs!).

• In a digraph, for any two vertices b and c, there is at most one edge from b to c.

• Similarly, in any undirected graph, there is at most one edge between two given vertices.

• No need to show edges for directed and undirected graphs, hence;

• Path1 = <c,e,d>. – just vertices• But labelling is required for multigraphs.

Simple Path

• A simple path is a path in which no vertex appears more than once (first and last may be same). Examples from Fig. 19.1:

• <a> an isolated node, a path with no edges,len=0

• <b,b> loop, first=last => cycle, length=1• <b,c,e,d>• <b,c,e,c> not simple path

An Example of a Simple-Path • In the following diagram the red edges show a

simple path of length 3 from a to e. The simple path is acde of length 3. We could also have a non-simple path acdabde of length 6.

• A simple path: no vertex appears more than once.

a

b c

d e

X

by

d

Z

cUses sequence notation

Cycle

• A path with at least one edge and with the first and last vertices the same is called a cycle.

• (19.7) Theorem: Suppose all the vertices of a loop free multigraph have even degree. Suppose deg.b > 0 for some vertex b. Then some cycle contains b. Note, zero is an even number and is neither positive nor negative.

• Note that unlike paths, any vertex of a cycle can be chosen as the start, so the start is sometimes not specified.

Connected Graph

• A undirected multigraph is connected if there is a path between any two vertices.

• A digraph is connected if making its edges undirected results in a connected multigraph.

digraph: not connected c

a

d

b

e

Fig. 19.1

d

c

b

a

Fig. 19.2 a

Connected graphs

•The graph in Fig 19.1 is not connected, the graphs in Fig 19.2a and Fig 19.3b are connected.

Connected Directed Graph1

•   There are three distinct forms of connectedness in simple directed graphs: weakly connected, unilaterally connected and strongly connected. 

A directed graph is strongly connected if every two vertices are reachable from each other. Unilaterally connected graphs need the concept of semi-path, which we do not cover.

1 7

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Path Simple Path? Cycle? Simple Cycle?

<6,5,2,4,3,2,1> no no no

<6,5,2,4> yes no no

<2,6,5,2,4,3,2> no yes no

<5,6,2,5> yes(book def) yes yes

<7> yes no no (length=0)

We have a path when no edge traversed more than once.A simple path from. v to w is a path from v to w with no repeated vertices. (first & last may be the same)<v0,e1,v1,e2,v2,…,vn-1,en,vn> where v0=v and vn=w

A cycle (or circuit) is a path of nonzero length from v to v with no repeated edges Must return to start.A simple cycle is a cycle is a non-zero sequence of vertices and edges, from v to v in which, except for the beginning and ending vertices that are both equal to v, there are no repeated vertices.

Graph Representation 1

• Adjacency matrix• Rows and columns

are labeled with ordered vertices write a 1 if there is an edge between the row vertex and the column vertex and 0 if no edge exists between them

v w x y

v 0 1 0 1

w 1 0 1 1

x 0 1 0 1

y 1 1 1 0

Graph Representation 2

• Incidence matrix– Label rows with vertices– Label columns with edges– 1 if an edge is incident to

a vertex, 0 otherwise

e f g h j

v 1 1 0 0 0

w 1 0 1 0 1

x 0 0 0 1 1

y 0 1 1 1 0

Three applications of Graphs

• The Konigsberg Bridge Problem

• The Celebrity Problem

• Route finding

• The states of river crossing puzzle (slides towards end of this lecture).

Celebrity problem uses a directed graph

• At a party, a celebrity is a person who everyone knows and who knows no one (except themselves).

• In graph theory terms what would be represented by nodes and vertices?

• Would the graph be directed or undirected?

Definition & Application of Euler Cycle

• An Euler cycle is a cycle in a graph that includes each edge exactly once. Examples: Designing and optimizing routes for refuse trucks, snow ploughs, or postmen. In all of these applications, every edge in a graph must be traversed at least once. For example, roads in a city must be completely traversed at least once in order to ensure that all rubbish is collected, all snow cleared, all post delivered. We seek to minimize total time, or the total distance or number of edges traversed. It may not always be possible to travel each edge exactly once and a near optimal solution may have to be used.

Euler Path

• An Euler path in a multigraph is a path that contains each edge exactly once. If the first and last vertices are the same then the Euler path is called an Euler cycle (or circuit). Example: G1 has Euler path from a to b

• a, c, d, e, b, d, a, b

a b

c d e

Graph G1

Euler Path

• a, c

a b

c d e

Euler Path

• a, c, d

a b

c d e

Euler Path

• a, c, d, e

a b

c d e

Euler Path

• a, c, d, e, b

a b

c d e

Euler Path

• a, c, d, e, b, d

a b

c d e

Euler Path

• a, c, d, e, b, d, a

a b

c d e

Euler Path

• a , c, d, e, b, d, a, b

a b

c d e

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

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Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

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Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

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i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycle

a b

c

d e

f g

ml

hk

on

j

i

Euler cycle: a,d,c,i,j,n,o,k,j,d,e,k,l,h,m,g,h,f,e,b,a

Euler Cycles and Paths

Has Euler cycle

Have Euler paths,but no Euler cycleFind Euler Paths. No repeating edgesIs Euler path unique?

No Euler cycleno Euler path

a b

c d e

The general theory from the Konigsberg Bridge Problem

• (19.8) Theorem. An undirected connected multigraph has an Euler circuit iff every vertex has even degree.

• Euler Circuit = Even Degree

• The proof is in two parts

• (1) LHS => RHS and (2) RHS => LHS

• This is called proof by mutual implication.

Example Proof

• Euler Circuit (EC)= Even Degree (ED)• (1) LHS => RHS • This says: Euler Circuit => is of even degree.• Assume that the multigraph has an EC as below:

v0 v1e0

v2e1

vnen

• Each vertex vi on the EC is preceded by an edge, and followed by an edge, and each of the two edges contributes 1 to the degree of vi. Since all the edges appear in the circuit the degree of each vi is even.

Proof(2)

• Euler Circuit(EC)= Even Degree(ED)• (2) RHS => LHS• Long proof. We will not cover the second part of

this proof. If interested, see course text.

A Simple Graph

• A graph with neither loops nor parallel edges is called a simple graph (we are talking about graphs not paths).

The graph G1 is a simple graph.

The graph G2 is not a simple graph

G1G2

A Complete Graph

• A complete graph on n vertices, Kn, is the simple graph (no loops or parallel edges) with n vertices in which there is an edge between every pair of distinct vertices. In general, the complete graph, Kn, is a simple graph with n vertices, each of which has degree n − 1.

K4

Example K4 n vertices and n(n − 1) / 2 edges4*(4-1)/2 = 6 edges

In any graph, the number of vertices with odd degree is even. In K4 each vertex is of

odd degree: 3+3+3+3=12 = 2 * 6 = 2|E|

Some Complete graphs with number of edges.

K1:0 K2:1 K3:3 K4:6

                                               

K5:10 K6:15 K7:21 K8:28

                

  

                

  

                

  

                

  

• A graph G=<V,E> is bipartite if there exists subsets V1 and V2 (either possibly empty) of V such that V1V2 = , V1V2 = V (i.e. a partition), and each edge in E is incident on one vertex in V1 and one vertex in V2 . A bipartite graph is a graph that does not contain any odd-length cycles.

v1

A bipartite graph

V1 ={v1 , v2 , v3}

V2 ={ v4, v5 ,}

V1V2 = ,

V1V2 =V

v2

v4

v3

v5

V1 V2

Bipartite Graph

• A graph G=<V,E> is bipartite if there exists subsets V1 and V2 (either possibly empty) of V such that V1V2 = , V1V2 = V (i.e. a partition), and each edge in E is incident on one vertex in V1 and one vertex in V2 . A bipartite graph is a graph that does not contain any odd-length cycles.

Bipartite Graph

V1 V2

A Complete Bipartite Graph• The complete bipartite on m and n vertices Km,n , is

the simple graph whose vertex set are partitioned into two sets V1 with m vertices and V2 with n vertices in which there is an edge between each pair of vertices, where v1 is in V1 and v2 is in V2 .

A complete bipartite graph K2,4

V1 V2

Application of bipartite graphs

• Bipartite graphs can be used to represent relations.

• Bipartite graphs can be used for mapping workers to possible jobs.

• Bipartite graphs are used in Petri nets to model processes. Nodes represent transitions (events-bars) and places (conditions-circles). The directed arcs describe which places are conditions for which transitions (arrows).

Application of bipartite graphs

The graph K3,3 is called the utility graph. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the non-planarity of K3,3.

Finding Bipartite graph

• Use two colours say blue & red. Such a colouring of the vertices of a bipartite graph means that the graph can be drawn with the red vertices on the left and the blue vertices on the right such that all edges go from left to right. Colour the first vertex blue, and then do a depth-first search of the graph. Whenever we discover a new, uncoloured vertex, colour it opposite that of its parent, since the same colour would cause a clash. If we ever find an edge where both vertices have been coloured identically, then the graph cannot be bipartite. Otherwise, this colouring will be a 2-colouring and can be constructed.

The edges are traversed in the order specified by the numbers above.

We will cover graph traversal and searching in the next lecture

Original Graph DFS

Depth First Search (DFS)

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4

56

7

Bipartite Example

• G1 is bipartite because it can be two coloured, blue for the centre and red for the rest.

• G2 (K3) is not bipartite because it cannot be two coloured. If we make V1 red then V2 and V3 must be blue, but V2 is connected to V3. Hence, blue to blue which is not allowed. It will be the same no matter where we start.

3

Finding a bipartition using parity

0 1

12

34

4

45

Odd parity

Even parity

Bipartite Example

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13

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6

41

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5

The graph is bipartite because every edge goes between an odd-numbered vertex (red) and an even numbered vertex (blue), giving

V = {1,3,5,7} U = {2,4,6,8}

Bipartite Example

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41

32

5

1

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K-colouring graphs

K-colouring graphs

A k-colouring partitions of the vertex set into k independent sets.

Sub-graph

e8

e10 e11

e5

v2

e8

e10 e11

e5

v2

Some subgraphs of G1

Graph G1

Sub-graph

Graph G The four possible subgraph G

v2

v1v1

e1

v2 v2

v1

v2

v1

e1

Graph G G1 G2 G3 G4

Bipartite Graph

• (19.10) Theorem. A path of a bipartite graph is of even length iff its ends are in the same partition.

v1

v2

v4

v3

v5

V1 V2

Bipartite Graph

• (19.11) Corollary. A connected graph is bipartite iff every cycle has even length e.g. look at the cycle <v3,v5,v2,v4,v3>.

v1

v2

v4

v3

v5

V1 V2

Bipartite Graph

• Another example of (19.11)

V1 V2

Complete Bipartite Graphs

K1,3 K2,3 K3,3

Example

Has the following graph got a Euler cycle. If it has what is it.

Solution: No Euler Cycle, vertex b has odd degree.

a

b

c d

g f

e

In general if you should test for existence, before trying to find Euler cycle

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, all vertices of even degree.

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5,v1>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5,v1,v4>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5,v1,v4,v3>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5,v1,v4,v3,v2>

v1 v2

v4

v3

v5

ExampleHas the following graph got a Euler cycle? If it has what is it.

Solution: Yes, <v1,v3,v5,v2,v4,v5,v1,v4,v3,v2,v1>

v1 v2

v4

v3

v5

Example

• Draw a graph with 4 edges and 4 vertices ,• having degrees 1,2,3,4.

• No such graph exists, because Handshaking Theorem(HT) says

• 2 * numberOfEdges(E) = sum(deg(V))• but• 2 * 4 =/= 1+2+3+4• Which is not consistent with HT

c

a

d

b

e

Fig. 19.1

Sequence Path? Simple Path? Cycle? Simple Cycle?

<c,e,d>

<b,c,e,c,e>

<e,c,e>

<b,b>

<b>

A Theory of Graphs1

A Theory of Graphs

mod* GRAPH { [ Vertex Edge ]

-- Notation from Chapter 19 Gries and Schneider-- We have two operations called s & t on edges.-- Both operations take an Edge and return a Vertex-- which can be either source or target op <_,_> : Vertex Vertex -> Edge op s : Edge -> Vertex op t : Edge -> Vertex

A Theory of Graphs -- General graph equations vars a b : Vertex eq s(< a , b >) = a . eq t(< a , b >) = b . -- A particular instance of graph ops e1 e2 e3 : -> Edge ops v1 v2 v3 : -> Vertex eq e1 = < v1 , v2 > . eq e2 = < v2 , v3 > . eq e3 = < v3 , v1 > . }

A Theory of Graphs

-- v1---e1---v2----e2---v3-- | | -- ----------e3--------------open GRAPH-- What is the start of e1 ?red s(e1) . v1-- Is e1 connect to e2?red t(e1) == s(e2) . true

A directed graph or digraph G1 c

a

d

b

e

d

c

b

a d

c

b

a

An undirected graph represented as a digraph

Fig. 19.1

Fig. 19.2bFig. 19.2 a

Sample Graph g2v1

v4

v3

v2v5

Graph g2

Sample Graph g3

v1

v4

v3

v2v5

Graph g2

v7

v6

a

Sample Graph Path

n1n2

d

n5

Graph with paths see CafeOBJ specification.

f

n3

b

c n4

e