Can you connect the dots as shown without taking your pen off the page or drawing the same line twice.

DECISION MATHS 1Graphs terminology

ObjectivesTo understand the key words: node, arc, order

of node, trail, path, closed trail, cycle, connected.

To know what Eulerian and semi-Eulerian graphs are and be able to identify them.

MathsAlgorithms Homework also Ex 1B p6 and Ex

1C p8Next Lesson: More Graph theory and

terminology.Leonhard Euler 1707-1783: Swiss

mathematician and general genius who invented graph theory!

Nodes/vertices and arcs/edges

The order/degree of a node is the number of arcs which meet at the node.

Hmmm...Given all arcs must start and end at a node

what can you say about the sum of the orders of all nodes in a graph?

• A walk is a sequence of arcs such that the end node of one arc is the start node of the next arc... If a walk does not travel an arc more than once then it is called a trail (or route).

• A path is a trail which passes through a node once only.

• A closed trail is a trail where the first node and the last node are the same.

A cycle is a closed trail where only the first and last nodes are the same.

So...A walk that only uses each of its arcs once is a

trail...A trail that has the same start and end node is a

closed trail.A path is a trail that only visits each node once.A cycle is a closed path (allowing the path to visit

its start node twice at the start and finish).So given this increasing specialism what do

you suppose a Hamiltonian Cycle is?A cycle that visits every node

http://www.cut-the-knot.org/Curriculum/Combinatorics/GraphPractice.shtml

• A connected graph is one where, for any two nodes, a path can be found connecting the two nodes.

• An Eulerian graph is a connected graph which has a closed trail containing every arc precisely once

• A semi-Eulerian graph is a connected graph which has a trail containing every arc precisely once (you can draw it without going back over a line or taking your pen off the page)

Some graphs are neither

Eulerian nor semi-Eulerian.

Eulerian

Not Eulerian

Eulerian

Semi-Eulerian

Eulerian GraphsAll nodes in an Eulerian

Graph have even order.This means the Graph is

fully traversable (there is a closed trail that uses every edge).

Recall a trail never repeats and edge.

You could draw it without taking your pen off the page and will start and end at the same vertex.

A semi-Eulerian Graph has two nodes of odd degree (one pair since odd nodes must come in pairs).

This means the graph is semi-traversable (there is a trail that uses every edge).

Note that the trail is not closed so the start and end points are different (the odd nodes).

You could draw it without taking your pen off the page but will start and end at different nodes.

Eulerian, Semi Eulerian OR Neither

Euler’s Problem

http://www.youtube.com/watch?v=CnU1ybtwgw8

Complete GraphsComplete graphs are graphs where every

node is connected to every other. They are denoted

Kn where n is the

number of nodes

More ThinkingWhen is a complete graph Eulerian?

Simple GraphsContains no loop arcs (connects a node to

itself) or multiple arcs between the same pair

A planar graph is one which can be drawn in a plane in such a way that arcs only meet at nodes.

Can you draw K4 without letting any lines cross?

Is K4 planar?

Can you draw K5 without letting any lines cross?

Is K5 planar?

Bipartite GraphsBipartite Graphs are graphs that have two

sets of vertices usually drawn in a vertical line

Bipartite GraphsVertices cannot be joined to other vertices in

that set

Bipartite GraphsVertices cannot be joined to other vertices in

that set

Complete Bipartite GraphsThe complete Bipartite Graphs have every

possible matching they are written Km,n where m,n are the number of nodes in each set

K4,4 K4,2

A Bipartite Graph Puzzle

TreesThe final piece of graph theory terminology

we need to know about are trees.Trees connect nodes but can have multiple

branches from each node unlike walks etc.

TreesA Tree

Spanning TreesA Spanning Tree includes every node

Applications - NetworksIt is possible for graphs to have weighted

arcs. These weights can mean distances, time or cost (or several other things)

13

11

12

13

2 31

2

3

1

2 5

Once a graph has weights it

can be called a network

Other ThingsIt is possible for graphs to be represented in a

matrix/table (1 represents an arc 0 or - no arc (you can also have weights in here too)

B

H

A

DC

E F

G01100010

10010001

10011000

01100100

00100110

00011001

10001001

01000110

H

G

F

E

D

C

B

A

HGFEDCBA

A

B

C

DE

01021

10100

01010

20111

10010

E

D

C

B

A

EDCBA