Directed graph: (V,E) such that

V is a set of vertices (or nodes)

E V V is a set of edges (or arcs)

a

b

c

d

e

in-degree = 2

out-degree = 1

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

E = { (b,a), (c,b),

(d,c),(e,d),(e,b),

(b,e),(e,a) }

Undirected graph: (V,E) such that

V is a set of vertices

E is a set of unordered edges

(i.e. unordered pairs of vertices)

sets of size 2

b

c

d

e

degree = 3

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

E = { {a,b}, {b,c},

{c,d},{d,e},{e,a},

{e,b} }

a

a is adjacent to b

• networks are graphs

• object hierarchies are graphs

• circuit layouts are graphs

complete graph on n vertices (Kn):

undirected graph containing an edge

between each pair of distinct vertices

Def

init

ion

...

bipartite graph: (V,E)

an undirected graph whose vertex set V can be partitioned

in two disjoint, nonempty sets V1 and V2 such that every

edge connects a vertex in V1 to a vertex in V2.

Def

init

ion

V1 V2 V1 V2

b

c

d

ea

1. Edge list: {a,b}, {b,c}, {c,d},{d,e},{e,a}, {e,b}

size:

adjacency time:

2. Adjacency list:

size:

adjacency time:

V vertices

E edges

Vertex Adjacencies

a b, e

b a, c, e

c b, d

d c, e

e a, b, d

3. Adjacency matrix:

b

c

d

ea

a b c d e

a 0 1 0 0 1

b 1 0 1 0 1

c 0 1 0 1 0

d 0 0 1 0 1

e 1 1 0 1 0

size:

adjacency time:

4. Incidence matrix:

1 2 3 4 5 6

a 1 1 0 0 0 0

b 0 1 1 0 1 0

c 0 0 0 0 1 1

d 0 0 0 1 0 1

e 1 0 1 1 0 0

12

34

56

size:

adjacency time:

A=

A2 ... A

Graph G1=(V1,E1) is isomorphic to G2=(V2,E2) iff

•(intuitive) can relabel vertices of G2 to get G1

•(mathematical) one-to-one function f: V1V2

with (a,b)E1 (f(a),f(b))E2

Def

init

ion

1 2

3 4

1 2

4 3~

isomorphic

Path (in an undirected graph):

a sequence of edges joined end-to-end

(usually denoted by their vertices, in order)

Simple path: path that does not use the same edge

more than once.

Def

init

ion

5 5

e.g. ({1,3},{3,4},{4,2},{2,5}) = <1,3,4,2,5>

Cycle: path that starts and ends at the same vertex:

Def’n

e.g. <5,2,4,3,1,2> but not <5,2,4,3,1,2,5>

Def’n

e.g. <1,2,4,3,1> or <5,2,4,3,1,2,5>

Def’n A path is Eulerian if it covers every edge in the

graph exactly once.

(if a graph has an Eulerian path, then you can

draw it without lifting your pencil)

2

1

4

3 5 6

7

8Eulerian path:

<1,2,3,4,5,6,7,8>

1 2

3 4

5

1 2

3 4

5

can replace

path with cycle

Theorem: Let G=(V,E) be an undirected graph. Then

EvVv

2)(degree

Why? every 1 edge connects 2 vertices

1 2

3 4

5vertex degree

1 4

2 4

3 3

4 3

5 2

sum = 4+4+3+3+2 = 16

# of edges = 8

Theorem: Let G=(V,E) be an undirected graph. Then

EvVv

2)(degree

Corollary: Always have an even # of vertices

with odd degree.

A graph is connected if there is a path

between every pair of distinct vertices.

Def

init

ion

1 2

3 4

5

1 2

3 4

5

Which graphs have

Eulerian

paths/cycles?

A graph has an Eulerian cycle iff it is connected

and every vertex has even degree.

A graph has an Eulerian path iff it is connected

and exactly two vertices have odd degree.

Th

eore

m

A brief sketch of the proof

Eulerian cycle all even degrees:

For a cycle, when enter vertex along one edge,

we must immediately leave along another edge:

1 2

3 4

5

Thus, for every vertex, #entrances = # exits.

degree(vertex) = #entrances + #exits = 2*#exits

is even.

For path, start and end vertices do

not to have even weight.

all even degrees Eulerian cycle

proof by algorithm:

1. start at an arbitrary vertex

2. wander around until stuck

1 2

3 4

5

we can only get stuck at the start vertex, since

wandering into a vertex reduces its available

edges by 1, making it odd so it has at least

one more available edge.

3. if edges are left over, do same with

a vertex with unused edge and splice

resulting cycles.

1 2

3 4

5

1 2

3 4

5

1 2

3 4

5

Def’n A path is Hamiltonian if it touches every vertex in

the graph exactly once.

A Hamiltonian circuit comes back to original vertex.

2

1

4

3 5 6

7

8Hamiltonian:

<1,2,3,4,5,6,7,8>

1 2

3 4

5

1 2

3 4

5

A graph has Hamiltonian circuit if

•it is connected

•has n3 vertices

•every vertex has degree n/2Th

eore

m

In general: Very hard to find Hamiltonian path.

Def

init

ion

A graph is planar if it can be drawn on a flat

piece of paper so that no two edges cross.

non-planar drawing

of a planar graph

a b

cd

planar drawing

of a planar graph

Useful for:

1. VLSI design (overlapping edges require

extra layers)

2. circuit design (cannot overlap wires on board)

through isomorphism

(by Kuratowski - prove for 0.3pts extra)

A graph is planar iff it does not contain a

subdivision of K5 or K3,3.Th

eore

m

V+F = E + C + 1

V = # of vertices

E = # of edges

F = # of faces

C = # of connected components

V=11 E=13 F=6 C=3

V=10 E=12 F=5 C=2

Def

init

ion

A tree is a connected, acyclic graph.

path between any two vertices

no cycles (i.e. circuits)

Def

init

ion

A forest is an acyclic graph.

Th

eore

m A graph is a tree iff there is exactly one path

between any two vertices.

Th

eore

m

Every tree with n vertices has n-1 edges.

rootlevel 0

level 1

level 2

level 3

a

b c

parent of c

parent of b

child of a

sibling of c

child of a

sibling of b

•All edges directed away from root.

•Depth (or level) is distance from root.

•Leaves are nodes with no children.

•Every node (except root) has 1 parent

(CS trees)

Def’n A binary tree is a rooted tree where every node

has at most 2 children.

root

Left

subtree

Right

subtree

In a full binary tree (complete), every internal node

has 2 children.

internal node

(i.e. non-leaf)

a

b e

c d f

g

Preorder:

1. visit the root

2. recursively traverse left subtree

3. recursively traverse right subtree

In order:

1. recursively traverse left subtree

2. visit the root

3. recursively traverse right subtree

Postorder:

1. recursively traverse left subtree

2. recursively traverse right subtree

3. visit the root

+

- *

2 4 +

1 3

Preorder: + - 2 * 4 + 1 3 (Lisp, Scheme)

In order: -2 + 4 * 1 + 3 (C, C++, Java)

Postorder: 2 - 4 1 3 + * + (HP, PS, Forth)

Binary Search Tree: left <= root <= right

•in order traversal gives sorted list

•easy to search