# Class Activity: Exploring MST Characteristics

We will be exploring creating graphs that showcase different aspects of MST
algorithms.

## Task 1:

True/False. A MST can sometimes contain a cycle. Justify your answer with one or
two sentences.

> Answer: False. By definition you can remove an edge from cycle and all
> vertices will still be connected. Therefor if the MST has a cycle in it then
> that would mean it has a redundant edge and is therefore not minimal.

## Task 2:

For the following graph, show the safe data structure after one call to
Borůvka's `AddAllSafeEdges` method. Draw the spanning tree F showing all of the
safe edges that were added to it after this first pass.

![Graph](./assets/graph.png)

## Task 3:

Create a graph with at least 6 nodes that is fully connected (a complete graph)
and has distinct edge weights that Boruvka's algorithm can solve in a single
call to `AddAllSafeEdges`. Submit a picture of the graph and the safe array/list
that is created by the call to `AddAllSafeEdges`.

## Task 4:

Create a graph with at least 6 vertices with distinct edge weights such that the
MST contains the edge with the largest weight. Submit a picture of this graph.

## Task 5:

Create a graph with at least 4 nodes that is fully connected (a complete graph)
and has distinct edge weights that Boruvka's algorithm exhibits its worst case
performance. First, define what the worst case perform is for a single call to
`AddAllSafeEdges`. Next, showcase this performance on your graph clearly showing
how the MST (F) looks after a single call to `AddAllSafeEdges` and showing the
contents of the array/list **safe**.

## Task 6:

Can you create a graph with at least 4 vertices with distinct edge weights such
that the MST does not contain the lightest edge? Explain your answer both
logically (1 to 2 sentences) and using the logic within Boruvka's
`AddAllSafeEdges`.

## Submission

Submit a picture of your answers to this canvas assignment. If you worked with
another person, acknowledge that by placing the names of all people you worked
with at the top of the first page of your submission.