5.7 Eulerizing Graphs

• Euler circuit and Euler path do not always exist .

• There are many graphs (in real life) that have more than 2 odd vertices.

• Instead of finding a route that travels along the edges of a graph and passes through each and every edge of the graph at least once, we want to find a a route that re-cross the fewest number of edges.

Eulerizing of a graph

• Eulerizing is the process of changing all odd vertices to even vertices by duplicating appropriate edges

Eulerizing GraphsEulerizing Graphs

First step is to identify the odd vertices.

Second step is to add duplicate copies of edges to create all even vertices

OPTIMAL ROUTE: duplicate the fewest number of edges

NOT an optimal route

illegal route

Eulerizing the following graphs

1)

2)

Semi-eulerizing of a graph

• Semi-eulerizing is the process of leaving 2 odd vertices on the graph unchanged and changing other odd vertices to even vertices.

Semi-eulerizing GraphsSemi-eulerizing Graphs

First step is to identify the odd vertices.

Second step is leave out 2 odd vertices and add duplicate copies of edges to create even vertices

OPTIMAL ROUTE:duplicate the fewest number of edges

NOT an optimal route

illegal route

Semi eulerizing the following graphs

1)

2)