Graph Theory

Graphs, Paths & Circuits

WHAT YOU WILL LEARN

• Graphs, paths and circuits• The Königsberg bridge problem

History This was developed by Leonhard Euler (pronounced

“oiler”) to study the Konigsberg Bridge problem. Konigsberg was situated on both banks of the Prigel

River in Eastern Prussia with a series of seven bridges connecting the banks via two islands.

The people of Konigsberg wanted to know if it was possible to cross all seven of the bridges without crossing any twice.

Definitions

A graph is a finite set of points called vertices (singular form is vertex) connected by line segments (not necessarily straight) called edges.

A loop is an edge that connects a vertex to itself.

A B

C D

Loop

Edge

Vertex

Not a vertex

Example: Map

The map shows the states that make up part of the Midwest states from Weather Underground, Inc. Construct a graph to show the states that share a common border.

Michigan

Ohio

Indiana

Kentucky

West Virginia

Solution

Each vertex will represent one of the states.

If two states share a common border, connect the respective vertices with an edge.

Solution (continued)

Michigan

Ohio

Indiana

Kentucky

West Virginia

MI

OHIN

KY WV

Definitions

The degree of a vertex is the number of edges that connect to that vertex.

A vertex with an even number of edges connected to it is an even vertex.

A vertex with an odd number of edges connected to it is an odd vertex.

MI, OH, and WV are even vertices

IN, KY are odd vertices

MI

OH

IN

KY WV

Definitions

A path is a sequence of adjacent vertices and edges connecting them.

C, D, A, B is an example of a path.

A circuit is a path that begins and ends at the same vertex.

A, C, B, D, A forms a circuit.

A B

C D E

A B

C D E

Definitions

A graph is connected if, for any two vertices in the graph, there is a path that connects them.

Examples of disconnected graphs.

A

B C

D

G H

JK

Definitions (continued)

A bridge is an edge that if removed from a connected graph would create a disconnected graph.

A

B C

D

bridge G H

JK

bridge

Select the graph with six vertices, a bridge, and a loop.

a.

c.

b.

d.

Select the graph with six vertices, a bridge, and a loop.

a.

c.

b.

d.

Represent the floor plan below as a graph where each vertex represents a room and each edge represents a doorway between rooms.

a.

c.

b.

d.

a.

c.

b.

d.

Draw a connected graph with all even vertices.

a.

c.

b.

d.

Draw a connected graph with all even vertices.

a.

c.

b.

d.