Exam 7: Graph Theory

The computers in an office area are connected as follows:

If a circuit of a graph visits each vertex exactly once, then it is called a(n) _______________.

The graph below has two odd-degree vertices, so no Euler circuit exists. Suppose we would like to find a route that backtracks as little as possible. What duplicate edge could we create to help us find this route?

If each vertex of a graph is connected to every other vertex by an edge, then it is called a(n) _______________ graph.

Which of the following vertices have degree 2?

One night you put a quarter into a slot machine and get back two quarters. The next night, you put these two quarters back in the machine one at a time, and each time two quarters came out. If you want to go home taking your winnings of $500, then what nights should this pattern be continued through?

Find an Euler circuit for the figure below.

Find an Euler circuit for the figure below.

Which of the following is not a practical situation where finding an Euler circuit may be important?

Use the cheapest link algorithm to find an approximate solution to the traveling salesman problem for the figure below.

Which of the following vertices have degree 2?

Use the nearest-neighbor algorithm starting at vertex A of the figure below to find an approximate solution to the traveling salesman problem.

According to Euler's theorem, the figure below has an Euler circuit.

How many Euler circuits starting from the vertex B are possible?

Suppose the edges of a certain graph represent phone lines that must be maintained, and the vertices represent junctions. A worker maintaining the lines would like to find an Euler circuit for this graph to increase ___________.

The edges in a certain graph represent roads, and the vertices represent intersections. We want to time the traffic signals at each intersection and not visit any intersection twice. Should we look for an Euler circuit or a Hamilton circuit?

In a complete binary tree of height H, the number of leaves equals _________.

Use the cheapest link algorithm to find an approximate solution to the traveling salesman problem for the figure below.

How many smaller circuits can be a part of a Hamilton circuit?

When applying the nearest-neighbor algorithm, if there are two or more vertices equally nearby, any one of them may be selected.