7.1. Preliminaries & Graph

TerminologyReference: A Course in Discrete Mathematical

Structures by LR Vermani

Budi Irmawati-NAIST’s student

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Isomorphic

α=VV’β=EE’

G (V,E)

x (x,y)

y

z

G’(V’,E’)

x’ =α(x) (x’,y’) = β(x,y)

y’ =α(y)

z’ =α(z)4 3

1 2

c

a

db

c

ba

d

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Graph that has multiple edges connecting the same vertices

Multigraph

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Directed graph that is multigraph

Directed Multigraph

graph which no loop and only have one edge between two vertices

Simple/Linear Graph

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Graph which each of its edges are weighted.

Ordered quadruple (V,E,f,g) or triple (V,E,f) or triple (V,E,f)

V : non-empty set of vertices

E : set/multiset of edges

f : function with domain V

g : map with domain E

Weighted graph

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VV1

(male)

a1

a2

⋮

an

V2 (female)

b1

b2

⋮

bn

VV1

(male)

a1

a2

⋮

an

V2 (female)

b1

b2

⋮

bn

Each vertex in V1 incident with an vertex in V2

Every vertex in V1 may not incident with a vertex in V2

Some vertex in V1 or V2 are isolated

sub graph 2

sub graph 1

Bipartite Graph

disjoint disjoint

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V1 = {a,b,g}

V2 = {c,d,e,f}

V = V1 V2

V1 V2 =

V1

V2

Fig. 7.34

Fig. 7.33

Bipartite Graph

not bipartite

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V = V1 V2

V1 V2 =

V1

V2

Every vertex in V1 adjacent every vertex in V2 and vice versa

Complete bipartite graph which V1 order m and V2 order n

The number of edges is mn instead of n(n-1)/2

Every vertex adjacent to every vertex in other subgraph

Km,n

Complete Bipartite Graph

Bipartite Graph

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Bipartite Graph

simple graph: two vertices – one edge

always bipartite

simple graph: at least three vertices (possibly)

trivial graph; three isolated vertices

only have one edge

have two edges, or

have three edges

alwaysbipartite

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Example 7.2Cube with 6 faces. Every cube are colored using 4 colors.Is it possible to stack the cubes to form column so that no color apears twice

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Example 7.2Cube with 6 faces. Every cube are colored using 4 colors.Is it possible to stack the cubes to form column so that no color apears twice

1 2 3 4

a R G W B

b G W B R

c W R B G

d G B W R

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Example 7.3How to cross a river for:5 couples1 boat for 3 personswife + husband if there a man

Possible crossing

one person

a couple

three ladies

three gents

couple & gent

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Example 7.4How to cross a river for:5 couples1 boat for at most 4 personswife + husband if there a man

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Example 7.5A man, a dog, a sheep, a basket of cabbageOnly can carry one item in crossing a riverCabbage cannot stay with a sheep and a dog cannot stay with a sheepHow to cross it ?

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Example 7.6

G’ = (V’,E’) where o(V’) = s ≥ 2.Vertex v1. G’ is connected graph.

v1 adjacent to v2, v2 to v3, and so on.Chain v1v2, v2v3, …, vs-1vs. at least s-1 edges in G’For o(V’) = 1, o(E’) ≥ o(V’) – 1

G = (V,E) is undirected graph with k componentso(V) = n, o(E) = mProve that m ≥ n - k

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Example 7.6

For k components, (E1,V1), (E2,V2), …, (Ek,Vk)

Ei Ej = for i j

E = E1 E2 … Ek and V = V1 V2 … Vk

o(E) = o(E1)+o(E2)+…+o(Ek) ≥ (o(V1)-1)+(o(V2)-1)+…+(o(Vk)-1)) = o(v1)+o(V2)+…+o(Vk) – k = o(V) - k