ch10 discrete mathematics direct graph

8/9/2019 Ch10 Discrete Mathematics Direct Graph

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Discrete Mathematics & Mathematical Reasoning

Chapter 10: Graphs

Kousha Etessami

U. of Edinburgh, UK

Kousha Etessami (U of Edinburgh UK) Discrete Mathematics (Chapter 6) 1 / 13


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Overview

Graphs and Graph Models

Graph Terminology and Special Types of Graphs

Representations of Graphs, and Graph

IsomorphismConnectivity

Euler and Hamiltonian Paths

Brief look at other topics like graph coloring



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What is a Graph?

Informally, agraphconsists of a non-empty set ofvertices(or nodes),and a setEofedgesthat connect (pairs of) nodes.

But different types of graphs (undirected, directed, simple, multigraph,

. . .) have different formal definitions, depending on what kinds of edges

are allowed.

This creates a lot of (often inconsistent) terminology.

Before formalizing, lets see some examples....

During this course, we focus almost exclusively on standard

(undirected) graphsanddirected graphs,which are our first two examples.



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A(simple undirected) graph:

LA

SF

NY

ED

Onlyundirectededges;at most one edgebetween any pair of distinct

nodes; andno loops(edges between a node and itself).

Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 4 / 13


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Adirected graph (digraph)(with loops):

LA

SF

NY

ED

Onlydirectededges;at most one directed edgefromany nodetoany

node; andloops are allowed.



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Asimple directed graph:

LA

SF

NY

ED

Onlydirectededges;at most one directed edgefrom any node to any

other node; andno loops allowed.



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Anundirected multigraph:

LA

SF

NY

ED

Onlyundirectededges; may containmultiple edgesbetween a pair of

nodes; butno loops.



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Anundirected pseudograph:

LA

SF

NY

ED

Onlyundirectededges; may containmultiple edgesbetween a pair of

nodes; andmay contain loops(even multiple loops on the same node).



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Adirected multigraph:

LA

SF

NY

ED

Onlydirectededges; may containmultiple edgesfrom one node to

another; butno loops allowed.

Warning: this differs slightly from the Rosen book terminology. The

books notion of directed multigraph would allow loops.



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Graph Terminology Zoo (ridiculous)

Type Edges Multi-Edges? Loops?

1. (simple undirected) graph Undirected No No

2. (undirected) multigraph Undirected Yes No

3. (undirected) pseudograph Undirected Yes Yes

4. directed graph Directed No Yes

5. simple directed graph Directed No No

6. directed multigraph Directed Yes No1

7. directed pseudograph Directed Yes Yes

8. mixed graph Both Yes Yes

We will focus on the two most standard types:

(1.) graphs (simple undirected), and(4.) directed graphs(also known asdigraphs).

1differs from book.



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Formal Defintion of Directed Graphs

Adirected graph(digraph),G= (V, E), consists of a non-empty set,

V, ofvertices(ornodes), and a setEV V ofdirected edges(orarcs). Each directed edge(u, v) Ehas astart(tail) vertexu, and a

end(head) vertexv.Note: a directed graphG= (V, E)is simply a set Vtogether with abinary relationEonV.



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Definition of (Undirected) Graphs

For a setV, let[V]k denote the set ofk-element subsets ofV.(Equivalently,[V]k is the set of all k-combinationsof V.)

A (simple,undirected)graph,G= (V, E), consists of a non-empty set

V ofvertices(or nodes), and a setE[V]

2

of (undirected)edges.Every edge{u, v} Ehas two distinct vertices u=vasendpoints,and such verticesuandvare then said to beadjacentin the graphG.

Note: the above definitions allow for infinite graphs, where |V|= .

In this course we will focus on finite graphs.



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Graph Models: Computer Networks

network where we careabout the number of links:we use a multigraph.

diagnostic self-links at datacenters: we use apseudograph.

network with multiple one-waylinks we use a directed!multi"graph.


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Applications of Graphs#$%$&'()*NG *+ A G&A,)

!labeled directed etc. ..."

graph theory can be used in modelling of:

+ocial networksCommunications networks

*nformation networks

+oftware design

(ransportation networks

iological networks

......


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Graph Models: +ocial Networks model social structures: relationships between people or

groups.

/ertices represent indi/iduals or organi0ations edgesrepresent relationships between them.

1seful graph models of social networks include:

friendship graphs- undirected graphs where twopeople are connected if they are friends !e.g. on2acebook"

collaboration graphs- undirected graphs where two

people are connected if they collaborate in a specificway

influence graphs- directed graphs where there is anedge from one person to another if the first person caninfluence the second


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Graph Models: +ocial NetworksExample: A friendshipgraph: two people areconnected if they are2acebook friends.

Example: An

influence graph


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*nformation Networks

*n a web graph web pages are representedby /ertices and links are represented by

directededges.*n a citation network:&esearch papers are represented by /ertices.

When paper A cites paper there is an edge

from the /erte3 representing paper A to the/erte3 representing paper .


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(ransportationGraphs

Graph models are e3tensi/ely used to studytransportation networks.

Airline networks can be modeled using

directed multigraphs where:airports are represented by /ertices

each flight is represented by a directed edgefrom the /erte3 representing the departureairport to the /erte3 representing thedestination airport

&oad networks modeled using graphs


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iological Applications

Graph models are used e3tensi/ely in manyareas of the biological science.

Niche overlap graphs model competition

between species in an ecosystem:

Example: nicheo/erlap graph for aforest ecosystem.


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Degree and neighborhood of a vertex

Definition 3. (he degreeof a vertex v in aundirected graph is the number of edgesincident with it. (he degree of the /erte3 vis

denoted by deg!v".

Definition 3. (he neighborhood !neighbor set"of a vertex vin a undirected graph, denoted

N(v)is the set of /ertices ad9acent to /.


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4egrees and Neighborhoods of%ertices

Example: What are the degrees andneighborhoods of the /ertices in the graphs Gand H

Solution: deg!a" ; 7 deg!b" ; N!b" ; =a, c, e, f > N!d" ; =c>.


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)andshaking (heorem

THEOREM 1 (Handshain! "emma#: *fG;!%$" is a undirected graph with medgesthen:

2m=vV deg(v)

Proof:

Each edge contributes twice to the degree count of allvertices. Hence, both the left-hand and right-hand sidesof this equation equal twice the number of edges. QED


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4egree of %ertices !continued"

Theorem $%An undirected graph has an e/ennumber of /ertices of odd degree.

Proof% ?et V5 be the /ertices of e/en degreeand V7 be the /ertices of odd degree in graph

G; !V E" with medges. (hen

must be

e/en sincedeg!v" ise/en foreach v@V5

e/en

must be e/en because 7mise/en and the sum of degrees

of /ertices of e/en degree ise/en.(hus since this is thesum of degrees of all /erticesof odd degree there must bean e/en number of them.


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)andshaking (heorem:$3amples

Example: )ow many edges are there in agraph with 56 /ertices each ha/ing degree si3

Solution: the sum of the degrees of the/ertices is 56 ; 6. (he handshaking

theorem says 7m; 6.+o the number of edges is m; B6.

Example: *f a graph has /ertices can each

/erte3 ha/e degree BSolution: (his is not possible by thehandshaking thorem because the sum of thedegrees of the /ertices B ; 5 is odd.


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4irected Graphs

Definition% (he indegreeof a vertex vdenoted deg!!v" is the number of edgesdirected into v. (he outdegreeof v denoteddeg"!v", is the number of edges directed out ofv. Note that a loop at a /erte3 contributes 5 to

both in-degree and out-degree.Example% *n the graph Gwe ha/e

deg!!a" ; 7 deg!!b" ; 7

deg!!c" ; B deg!!d" ; 7deg!!e" ; B deg!!f" ; 6.


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4irected Graphs !continued"

Theorem 3: ?et G # !V, E"be a directed graph.(hen:

Proof: (he first sum counts the number ofoutgoing edges o/er all /ertices and the second

sum counts the number of incoming edges o/erall /ertices. oth sums must be D$D.


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+pecial (ypes of Graphs: CompleteGraphs

A complete graph on n vertices denoted by is the simple graph that contains e3actly oneedge between each pair of distinct /ertices.

Kn


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+pecial (ypes of Graphs: Cycles

A c$cle for nE B consists of n/ertices v5v7,% ,vn and edges =v5, v7>, =v7, vB>,% ,=vn5, vn>, =vn, v5>&

CnCnCn


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+pecial (ypes of +imple

Graphs: n-CubesAn ndimensional h$percube or ncube is agraph with /ertices representing all bitstrings of length n where there is an edge

between two /ertices if and only if they differ ine3actly one bit position.

2n


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ipartite Graphs

Definition%

An eFui/alent definition of a bipartite graph isone where it is possible to colorthe /erticeseither red or blue so that no two ad9acent/ertices are the same color.

Gisbipartite

His notbipartite: if wecolor aredthen itsneighbors fandbmust be blue.ut f and b aread9acent.


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ipartite Graphs !continued"

Example: +how that is bipartite.Solution: ,artition the /erte3 set into V5 ;=v5 vB v> and V7 ; =v7 v:

Example: +how that ' is not bipartite.

Solution: *f we partition /ertices of 'B into

two nonempty sets one set must contain two/ertices. ut e/ery /erte3 is connected to e/eryother. +o the two /ertices in the same partitionare connected. )ence 'B is not bipartite.

C6


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Complete ipartite Graphs

Definition% A complete bipartite graphis a graph that has its /erte3 set partitionedinto two subsets V5 of si0e mand V7 of si0e nsuch that there is an edge from e/ery /erte3 in

V5 to e/ery /erte3 in V7&

Examples%

Km,n


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+ubgraphs

Definition% A subgraphof a graph G;!VE" is a graph !*+" where

and . A subgraph Hof Gis a propersubgraphof Gif H G&Example: here is and one of its !proper"subgraphs:

WV

FEFE

K5


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*nduced +ubgraphs

Definition%?et G; !V E" be a graph. (hesubgraph induced by a subset * of the /erte3

set Vis the graph ); !*+" whose edge

set + contains an edge in E if and only if bothendpoints are in *&

Example: )ere is - and itsinduced

subgraphinduced by *; =a,b,c,e>.


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ipartite Graphs and Matchings

ipartite graphs used e3tensi/ely in apps

in/ol/ing matching elements of two sets:.ob assignments- /ertices represent the 9obsand the employees edges link employees with9obs they are Fualified for. Ma3imi0e H ofemployees matched to 9obs.

/arriage0dating- /ertices represent men Iwomen and edges link a man I woman if theyare acceptable to each other as partners.

B


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Bipartite graphs

Abipartite graphis a (undirected) graphG= (V,E)whosevertices can be partitioned into two disjoint sets (V1,V2), withV1 V2= andV1 V2=V, such that for every edgeeE,e={u, v}such thatuV1 andvV2. In other words, every

edge connects a vertex inV1 with a vertex in V2.

Equivalently, a graph isbipartiteif and only ifit is possible to

color each vertex red or blue such that no two adjacent vertices

are the same color.


E l f Bi i G h


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Example of a Bipartite Graph

V1 V2


M hi i Bi i G h


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Matchings in Bipartite Graphs

Amatching,M, in a graph,G= (V,E), is a subset of edges,ME, such that there does not exist two distinct edges inMthat are incident on the same vertex. In other words, if

{u, v}, {w, z} M, then either{u, v}= {w, z}or{u, v} {w, z}= .

Amaximum matchingin graphGis a matching inGwith the

maximum possible number of edges.


P f / l hi


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Perfect/complete matchings

For a graphG= (V,E), we say that a subset of edges, WE,coversa subset of vertices,A V, if for all verticesuA,there exists an edgeeW, such thateis incident onu, i.e.,such thate={u, v}, for some vertexv.

In a bipartite graphG= (V,E)with bipartition(V1,V2), acomplete matchingwith respect toV1, is a matchingM

Ethat coversV1, and aperfect matchingis a matching,M

E,that coversV.

Question: When does a bipartite graph have a perfectmatching?



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Halls Marriage Theorem

For a bipartite graphG= (V,E), with bipartition(V1,V2), there

exists a matchingMEthat coversV1 if and only iffor allSV1,|S| |N(S)|.

Proof: ForG= (V,E), withA V, letNG(A)denote theneighbors ofAinG.

First,only ifdirection: Suppose there is a matchingM inGthatcoversV1. We show thatSV1,|S| |NG(S)|. Suppose, forcontradiction, that there is a subsetSV1 such that|S| >|NG(S)|. Then no matchingMcould possibly coversS,because there arent enough neighborsN

G(S

). Done.

Theifdirection of the proof is harder...


f f H ll Th ti d


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proof of Halls Theorem, continued...

If direction: SupposeSV1,|S| |NG(S)|. Then we prove amatchingMexists which coversV1, byinductionon the size

|V1|.

Base case: |V1|= 1. Since|V1| |NG(V1)|, there must be an

edge covering the vertexu inV1={u}.

Inductive step: Suppose (byinductive hypothesis) that the

claim holds for bipartite graphsG with|V1|=jk. Suppose

|V1|=k+ 1.


f f H ll Th ( ti d)


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proof of Halls Theorem (continued)

Case 1: Suppose that for every nonempty strict subsetSV1,we have|S| |NG(S)| 1. Take any{u, v} E, withuV1.

Removeuandv(and the edges incident on them) from G. Call

the resulting bipartite graphG, with bipartition

(V1 {u},V2 {v}).By the induction hypothesis, there must exist a matching M inG that coversV1 {u}, because for every subset SV1 {u},NG(S) NG(S) {v}, and thus|NG(S)| |NG(S)| 1 |S|.But thenM=M {{u, v}}is a matching inGwhich coversV1.



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Case 2: Suppose, on the contrary, that there exists a nonempty

strict subsetSV1 with|S|=|NG(S)|.

Any matching that coversV1 must matchStoNG(S).By the induction hypothesis, there is a matchingM covering S

on the bipartite subgraphG ofGinduced byS NG(S). Andfurthermore, the bipartite subgraphG ofGinduced by

(V1 S) (V2 NG(S))also satisfies the condition, andcontains a matchingM that covers(V1 S). This is because ifA V1 Shas|A| >|NG(A)|, this implies|A S| >|NG(A S)|, which violates the assumption about G.

LettingM=M M,Mdefines a matching in Gthat covers

V1.


M M t hi


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More on Matchings

CorollaryA bipartite graph G= (V,E)with bipartition(V1,V2)has aperfectmatching if and only if|V1|= |V2|andS V1,|S| |NG(S)|.

Unfortunately, the proof we have given isnot constructive

enough: it doesnt yield an (efficient) algorithm to compute amaximum matching in a bipartite graph.

An alternative proof of Halls theorem (which we do not give)

based onalternating pathsandaugmenting paths, is

constructive & yields an efficient (polynomial time) algorithm forcomputing a maximum matching.



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New Graphs from 8ldDefinition: (he unionof two simple graphsG5# !V5, E5"and G7# !V7, E7"is the

simple graph with /erte3 set V5JV7and edge

set E5JE7. (he union of G5 and G7is denotedby G5J G7.

Example:


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&epresenting Graphs: Ad9acency ?ists

Definition: An ad1acenc$ list represents a

graph !with no multiple edges" by specifyingthe /ertices that are ad9acent to each /erte3.

Example:

Example:


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&epresentation of Graphs:

Ad9acency MatricesDefinition: +uppose that G; !V E" is a simplegraph where DVD ; n. Arbitrarily list the /erticesof Gas v(5 v(7 K v(n.

(he ad1acenc$ matrix' of G with respect tothis listing of /ertices is the n 2n6-5 matri3with its !i1"th entry ; 5 when v(i and v(1aread9acent and ;6 when they are not ad9acent.

*n other words: and:A=[aij]


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Ad9acency Matrices !continued"

Example: 3he vertex ordering isisa b c d.

(ote: (he ad9acency matri3 of an undirected graph is symmetric:

Also since there are no loops each diagonal entry is 4ero5

A sparsegraph has fewedges relati/e to thenumber of possibleedges. +parse graphsare more efficient torepresent using an

ad9acency list than anad9acency matri3. utfor a densegraph anad9acency matri3 isoften preferable.

aij=aji,i , j

aii=0 ,i


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Ad9acency Matrices !continued"

Ad9acency matrices can also be used torepresent graphs with loops and multi-edges.

When multiple edges connect /ertices viandv1 !or if multiple loops present at the same/erte3" the !i1"th entry eFuals the number of

edges connecting the pair of /ertices.Example: Ad9acency matri3 of a pseudographusing /erte3 ordering a b c d5


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Adjacency Matrices (continued)

Adjacency matrices can represent directed

graphs in exactly the same way. The matrix Afor a directed graph G= (V, E) has a 1 in its(i,j)th position if there is an edge from vi tovj, where v1, v2, vnis a list of the !ertices. "n other words,

#ote$ the adjacency matrix for a directedgraph need not %e symmetric.

aij=1 if (i , j)Eaij=0 if (i , j)E


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Isomorphism of Graphs

Definition: Two (undirected) graphs

G1 = (V1, E1)and G2= (V2, E2)areisomorphicif there is a bijection, ,

with the property that for all ertices

if and only if

!uch a function f is called an isomorphism"

Intuitiely, isomorphic graphs are #T$% !&'%,ecept for #renamed ertices"

{f(a) , f(b)}E2

a ,bV1

f :V1V

2

{a ,b}E1


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*somorphism of Graphs !cont&"

Example: +how that the graphs G;!V E" and

H; !* +" are isomorphic.

Solution: (he function fwith f!u5" ; v5f!u7" ; v


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*t is difficult to determine whether two graphsare isomorphic by brute force: there are nbi9ections between /ertices of two n-/erte3graphs.

8ften we can show two graphs are not

isomorphic by finding a property that only oneof the two graphs has. +uch a property is calledgraph invariant: e.g. number of /ertices of gi/en degree the

degree seFuence !list ofthe degrees" .....


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Example: Are these graphs are isomorphic

Solution: No +ince deg!a" ; 7 in G amustcorrespond to t ux or$,since these are the/ertices of degree 7 in ). ut each of these

/ertices is ad9acent to another /erte3 of degree7 in H which is not true for ain G. +o G and) can not be isomorphic.


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Example: 4etermine whether thesetwo graphs are isomorphic.

Solution: (he function fis defined

by: f!u5" ; v f!u7" ; vB f!uB" ;v


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Algorithms for Graph *somorphism

(he best algorithms known for determiningwhether two graphs are isomorphic ha/ee3ponential worst-case time comple3ity !in

the number of /ertices of the graphs".)owe/er there are algorithms with good

time comple3ity in many practical cases.

+ee e.g. a publicly a/ailable software calledNA1(' for graph isomorphism.


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Applications of Graph *somorphism

(he Fuestion whether graphs are isomorphicplays an important role in applications of graphtheory. 2or e3ample:

Chemists use molecular graphs to model chemical

compounds. %ertices represent atoms and edgesrepresent chemical bonds. When a newcompound is synthesi0ed a database ofmolecular graphs is checked to determine

whether the new compound is isomorphic to thegraph of an already known one.

Section Summary


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Section Summary

Paths

Connectedness in Undirected Graphs

(strong) Connectedness in Directed Graphs


Paths (in undirected graphs)


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Paths (in undirected graphs)Informally, apathis a sequence of edges connecting vertices.

Formally:

Definition:For an undirected graphG= (V, E), an integern0, and verticesu, vV, apath (or walk) of lengthnfromutov inGis a sequence:

x0,e1, x1, e2, . . . , xn1,en, xn

of interleaved verticesxjVand edgeseiE,such thatx0=uandxn=v, and such thatei={xi1, xi} E foralli {1, . . . , n}.

Such a pathstartsatuandendsatv. A path of lengthn1 iscalled acircuit(orcycle) ifn1 and the path starts and ends atthe same vertex, i.e.,u=v.

A path or circuit is calledsimpleif it does not contain the same

edge more than once.


More on paths


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More on pathsWhenG= (V,E)is asimple undirected grapha pathx0, e1, . . . , en, xnis determineduniquelyby the sequence of

verticesx0

, x1

, . . . , xn

. So, for simple undirected graphswe can

denote a path by its sequence of verticesx0, x1, . . . , xn.

Note 1: The word simple is overloaded. Dont confuse a

simpleundirected graph with asimplepath. There can be a

simple path in a non-simple graph, and a non-simple path in a

simple graph.

Note 2: The terms path and simple path used in Rosens

book are not entirely standard. Other books use the terms

walkandtrailto denote path and simple path, respectively.

Furthermore, others use path itself to mean a walk that doesntre-visit any vertex, except possibly the first and last in case it is

a circuit. To stick to Rosens terminology, we shall use the

non-standardtermtidy pathto refer to such a walk.


Example


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ExampleHere is a simple undirected graph:

a b c

d g f

d,a, b, c, f is a simple (and tidy) path of length 4.

d,g, c, b,a,d is a simple (and tidy) circuit of length 5.

a, b,g, f is nota path, because{b,g}is not an edge.

d,a, b, c, f, b,a,g isa path, but itis nota simple path,because the edge{a,b}occurs twice in it.c,g,a, d, g, fis a simple path, but itis nota tidy path,

because vertexgoccurs twice in it.Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 4 / 20

Example: an acquantance graph


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Example: an acquantance graph

Alice

Bob

Kathy

David

Ellen

Fred


Paths in directed graphs (same definitions)


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Paths in directed graphs (same definitions)

Definition:For an directed graphG= (V, E), an integern0,and verticesu, vV, apath (or walk) of lengthnfromutov in

Gis a sequence of vertices and edges x0,e1, x1, e2, . . . , xn,en,such thatx0 =uandxn=v, and such thatei= (xi1, xi)E foralli {1, . . . , n}.

When there are no multi-edges in the directed graph G, the path

can be denoted (uniquely) by its vertex sequencex0, x1, . . . , xn.

A path of lengthn1 is called acircuit(orcycle) if the pathstarts and ends at the same vertex, i.e.,u=v.

A path or circuit is calledsimpleif it does not contain the same

edge more than once. (And we call ittidyif it does not containthe same vertex more than once, except possibly the first and

last in caseu=vand the path is a circuit (cycle).)


Connectness in undirected graphs


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Definition:An undirected graphG= (V, E)is calledconnected,if there is a path between every pair of distinct vertices.

It is calleddisconnnectedotherwise.

a

b

c

d e

f

This graph is connected




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Definition:An undirected graphG= (V, E)is calledconnected,if there is a path between every pair of distinct vertices.

It is calleddisconnnectedotherwise.

a

b

c

d e

f

This graph is connected

a

b

c

d e

f

This graph isnotconnected



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Proposition

There is always a simple, and tidy, path between any pair ofverticesu, vof a connected undirected graphG.

Proof: By definition of connectedness, for every pair of vertices

u, v, there must exist ashortestpathx0,e1, x1, . . . , en, xn inG

such thatx0 =uandxn=v.Suppose this path is not tidy, andn1. (Ifn=0, theProposition is trivial.) Thenxj=xkfor some 0j


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connected components of undirected graphs

Definition:Aconnected componentH= (V,E)of a graph

G= (V,E)is amaximalconnected subgraph ofG, meaningHis connected andV V andE E, butHis not a propersubgraph of a larger connected subgraph RofG.

a

b

c

d e

f

g h

This graph,G= (V, E), has 3 connected components.

(It is thus a disconnected graph.)One connected component ofGisH1= (V

1, E

1),whereV1={d,a,b}andE

1 ={{d, a}, {d, b}}.


Connectedness in directed graphs


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Connectedness in directed graphs

Definition:A directed graphG= (V,E)is calledstronglyconnected, if for every pair of vertices uandv inV, there is a

(directed) path fromutov,anda directed path fromv tou.

(G= (V, E)isweakly connectedif there is a path between

every pair of vertices inVin the underlying undirected graph(meaning when we ignore the direction of edges in E.)

Astrongly connected component (SCC)of a directed graph G,

is a maximal strongly connected subgraphHofGwhich is not

contained in a larger strongly connected subgraph ofG.


Example


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Example

a

b

c

d e

f

This digraph,G, isnotstrongly connected, because, for

example, there is no directed path frombtoc.

Question: what are the strongly connected components (SCCs)ofG?


Example


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Example

a

b

c

d e

f

This digraph,G, isnotstrongly connected, because, for

example, there is no directed path frombtoc.

Question: what are the strongly connected components (SCCs)ofG?

One strongly connected component (SCC) ofGisH1 = (V

1, E

1),whereV1={d,a,b}andE

1 ={(d,a),(a, b),(b,d)}.

Another SCC ofGisH2= (V

2,E

2), whereV

2={e, c, f}andE2={(e, c),(c, f),(f,e)}.

There are no other SCCs in G.Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 11 / 20

Directed Acyclic Graphs


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Directed Acyclic Graphs

ADirected Acyclic Graph (DAG), is a directed graph that

contains no circuits or loops.

Example:

a

b

c

d

e

f

a

b

c

d

e

f

This is a DAG This is NOT a DAG



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Euler Paths and Euler Circuits

Hamiltonian Paths and Hamiltonian Circuits


The Knigsberg Bridge Problem


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e gsbe g dge ob eLeonard Euler (1707-1783) was asked to solve the following:

Question: Can you start a walk somewhere in Knigsberg, walk

across each of the 7 bridgesexactly once, and end up back

where you started from?


The Knigsberg Bridge Problem


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g g gLeonard Euler (1707-1783) was asked to solve the following:

Question: Can you start a walk somewhere in Knigsberg, walk

across each of the 7 bridgesexactly once, and end up back

where you started from?

Euler (in 1736) used graph theory to answer this question.

A

B

C

D


Euler paths and Euler Circuits


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p

Recall that an (undirected)multigraphdoes not have any loops,

but can have multiple edges between the same pair of vertices.

Definition: AnEuler pathin a multigraphGis a simple path that

contains every edge ofG.

(So, every edge occurs exactly once in the path.)

AnEuler circuitin an multigraphGis a simple circuit thatcontains every edge ofG.

(So, every edge occurs exactly once in the circuit.)

Question: Is there a simple criterion for determining whether a

multigraph Ghas an Euler path (an Euler circuit)?



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Eulers Theorem


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Eulers Theorem (1736)

A connected undirected multigraph with at least two vertices hasan Euler circuit if and only if each of its vertices has even

degree.

Proof: Only if direction:Suppose a multigraphG= (V,E)has

an Euler circuit,x0e1x1e2. . . emxm, wherex0 =xm=u.For every vertexvV,v=u, each time we entervvia an edgeei, we must leavevvia a different edgeei+1. So, in total, since

we see all edges incident tovexactly once, all such verticesv

must have even degree.

Likewise, the initial (and final) vertexu=x0=xm, must alsohave even degree, because the edgese1andempair up in the

same way.


Proof of Eulers Theorem (continued)


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( )

The(harder)if direction:SupposeG= (V,E)is connectedand every vertex inVhas even degree.

We give aconstructive proofthat, given such a multigraphG,shows how to construct an Euler circuit (efficiently).

Start a walk at any vertexv, never re-using an edge, walking

for as long as possible until you can not do so any more.

Claim: Such a walk (simple path),w1, must end at the vertexv

where it started (i.e., it must be a circuit).

Reason: For any vertexzother thanv, whenever the walk

enterszvia an edge, there must be an odd number of edges

incident tozremaining. Note: zero is not an odd number! After

leavingz, there must be an even number of edges ofzremaining.

If the simple circuitw1 covers every edge of G, we are done.

If not, .....


Proof of Eulers theorem (final part)


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( p )Note that every vertex has even degree remaining after the

edges of the simple circuitw1 are removed.

If the simple circuitw1 does not cover every edge ofG, sinceGis connected, there must be some vertex x on the circuitw1which is incident to an edge not inw1. So,w1=w

1xw

1

We start a new walk at the vertex x, on the remaining graph

without the edges ofw1. This yields a new circuitw2 that must

start and end atx

.We can then then splice w2 insidew1 (at the point wherex

occurs) in order to get a new longer Euler circuit: w1w2w

1 .

We can do this same process repeatedly until there are no

edges remaining.

Note: this also yields a reasonably efficient algorithm for

computing an Euler circuit in a connected multigraph where

every vertex has even degree.


Eulers theorem for paths


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p

Eulers Theorem for pathsA connected undirected multigraphGhas an Euler path which is

notan Euler circuit if and only if Ghas exactly two vertices of

odd degree.

The proof is very similar to the case of Euler circuits: just start

the initial walk at one of the vertices of odd degree.

The proof is thus similarly constructive, and yields an efficient

algorithm to construct an Euler path, if one exists.


Hamiltonian Paths


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Definition: AHamiltonian pathin a (undirected) graph Gis a

simple path that visits every vertex exactly once. (In other

words, it is a tidy path that visits every vertex.)

AHamiltonian circuitin a (undirected) graphGis a simple circuit

that passes through every vertex exactly once (except for the

common start and end vertex, which is seen exactly twice).


(simple undirected) graph has a Hamiltonian path, or

Hamiltonian circuit?


Hamiltonian Paths


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Definition: AHamiltonian pathin a (undirected) graph Gis a

simple path that visits every vertex exactly once. (In other

words, it is a tidy path that visits every vertex.)

AHamiltonian circuitin a (undirected) graphGis a simple circuit

that passes through every vertex exactly once (except for the

common start and end vertex, which is seen exactly twice).


(simple undirected) graph has a Hamiltonian path, or

Hamiltonian circuit?

Answer: No. Nobody knows any efficient algorithm fordetermining whether a given (arbitrary) graphGhas a

Hamiltonian path/circuit. The problem is NP-complete.


More on Hamiltonian paths/circuits


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p

There aresufficientcriteria that guarantee existence of a

Hamiltonian circuit. For example:

Ores TheoremEvery simple undirected graph,G= (V,E), withn3 vertices,in whichdeg(u) + deg(v)nfor every two non-adjacentverticesuandv inV, has a Hamiltonian circuit.

Corollary (Diracs Theorem)

Every simple undirected graph,G= (V,E), withn3 vertices,in whichdeg(u)n/2 for all verticesuV, has a Hamiltonian

circuit.

We will NOT prove these theorems, and we will NOT

ch10 discrete mathematics direct graph

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