Exam 1: Urban Services

Describe the two conditions for a graph to have an Euler circuit.

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -

Every graph with an Euler circuit has only vertices with even valences.

Use the following to answer Questions: Identify an Euler circuit on the following graphs by numbering the sequence of edges in the order traveled. -

In order to eulerize the graph below, give the fewest number of edges that need to be duplicated.

In order to eulerize the graph below, give the fewest number of edges that need to be added or duplicated.

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -

Which of the graphs shown below gives the best eulerization of the given graph? (In the graphs below, added edges are denoted with zigzag lines.)

In the graph below, find the largest number of paths from A to G that do not have any edges in common.

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -

For which of the two situations below is it desirable to find an Euler circuit or an efficient eulerization of a graph? I. A street department employee must check the traffic signals at each intersection in a downtown area to be certain they are working. II) An employee of a power company reads the electric meters outside each house along the streets in a residential area.

Why would a city street department want its snowplow operator's path to follow an Euler circuit if possible?

What is the sum of all valences of the graph shown below?

Consider the path represented by the numbered sequences of edges in the graph below. Which statement is true?

Give three real-world applications in which a worker would want to find an Euler circuit on a street network.

Consider the path represented by the numbered sequences of edges in the graph below. Which path represents an Euler circuit?

After a major natural disaster, such as a flood, hurricane, or tornado, many tasks need to be completed as efficiently as possible. For which situation below would finding an Euler circuit or an efficient eulerization of a graph be the appropriate mathematical technique to apply?

Draw a graph where every vertex has a valence of at least 2 but removing a single edge disconnects the graph.

Use the following to answer Questions: Identify an Euler circuit on the following graphs by numbering the sequence of edges in the order traveled. -

Use the following to answer Questions: Add wiggly edges to find an efficient eulerization of the following graphs. -