Collapse all questions into a single README

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-Given the graph shown below, answer the following questions:
-
-1. Illustrate the sequence of vertices of this graph visited using depth-first search traversal starting at vertex `g`.
-1. Illustrate the sequence of vertices of this graph visited using breadth-first search traversal starting at vertex `b`.
-1. Illustrate adjacency list representation and adjacency matrix representation, respectively, for this graph.
-  * What are the advantages and disadvantages of those two representations?
-1. Describe an algorithm to find in the graph a path illustrated below that goes through every edge exactly once in each direction.
-
-
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-# Depth First Traversal
-
-Order: g, h, o, p, l, k, n, i, m, j, f, c, d, b, a, e
-
-1. [g]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-2. [g, h]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-3. [g, h, o]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(*)---(p)
-```
-
-4. [g, h, o, p]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(*)---(*)
-```
-
-5. [g, h, o, p, l]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (*)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(*)---(*)
-```
-
-6. [g, h, o, p, l, k]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(*)---(*)
-```
-
-7. [g, h, o, p, l, k, n]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(m)  \(*)---(*)---(*)
-```
-
-8. [g, h, o, p, l, k, n, i]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(j)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(m)  \(*)---(*)---(*)
-```
-
-9. [g, h, o, p, l, k, n, i, m]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(j)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-10. [g, h, o, p, l, k, n, i, m, j]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-11. [g, h, o, p, l, k, n, i, m, j, f]
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-12. [g, h, o, p, l, k, n, i, m, j, f, c]
-```plaintext
-(a)---(b)---(*)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-13. [g, h, o, p, l, k, n, i, m, j, f, c, d]
-```plaintext
-(a)---(b)---(*)---(*)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-14. [g, h, o, p, l, k, n, i, m, j, f, c, d, b]
-```plaintext
-(a)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-15. [g, h, o, p, l, k, n, i, m, j, f, c, d, b, a]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-16. [g, h, o, p, l, k, n, i, m, j, f, c, d, b, a, e]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-# Breadth First Traversal
-
-Order: [b, a, f, c, e, j, d, i, g, m, n, h, k, o, p, l]
-
-1. [b]
-```plaintext
-(a)---(*)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-2. [b, a]
-```plaintext
-(*)---(*)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-3. [b, a, f]
-```plaintext
-(*)---(*)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-4. [b, a, f, c]
-```plaintext
-(*)---(*)---(*)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-5. [b, a, f, c, e]
-```plaintext
-(*)---(*)---(*)---(d)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-6. [b, a, f, c, e, j]
-```plaintext
-(*)---(*)---(*)---(d)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-7. [b, a, f, c, e, j, d]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-8. [b, a, f, c, e, j, d, i]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-9. [b, a, f, c, e, j, d, i, g]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-10. [b, a, f, c, e, j, d, i, g, m]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(n)---(o)---(p)
-```
-
-11. [b, a, f, c, e, j, d, i, g, m, n]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(o)---(p)
-```
-
-12. [b, a, f, c, e, j, d, i, g, m, n, h]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(o)---(p)
-```
-
-13. [b, a, f, c, e, j, d, i, g, m, n, h, k]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(o)---(p)
-```
-
-14. [b, a, f, c, e, j, d, i, g, m, n, h, k, o]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(p)
-```
-
-15. [b, a, f, c, e, j, d, i, g, m, n, h, k, o, p]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (l)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-16. [b, a, f, c, e, j, d, i, g, m, n, h, k, o, p, l]
-```plaintext
-(*)---(*)---(*)---(*)
- | \       /     /
- |  \     /     /
-(*)  \(*)/  (*)/--(*)
- |     |   / |    /
- |     |  /  |   /
-(*)---(*)/  (*) / (*)
- | \         | /   |
- |  \        |/    |
-(*)  \(*)---(*)---(*)
-```
-
-# Adjacency List
-
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-
-(a) -> [b, e, f]
-(b) -> [a, f]
-(c) -> [b, d, f]
-(d) -> [c, g]
-(e) -> [a, i]
-(f) -> [a, c, j]
-(g) -> [d, h, j, k]
-(h) -> [g, o]
-(i) -> [e, j, m, n]
-(j) -> [f, g, i]
-(k) -> [g, o]
-(l) -> [p]
-(m) -> [i]
-(n) -> [i, o]
-(o) -> [k, n, p]
-(p) -> [l, o]
-```
-
-Good:
-
-* Space efficient because no space is wasted for edges that do not exist.
-
-Bad:
-
-* A lookup to determine if two vertexes are connected requires a linear time lookup due to the size of the list for a single edge. `O(n)`.
-
-# Adjacency Matrix
-
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-```plaintext
------------------------------------
-| |a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|
-|a|0|1|0|0|1|1|0|0|0|0|0|0|0|0|0|0|
-|b|1|0|1|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|c|0|1|0|1|0|1|0|0|0|0|0|0|0|0|0|0|
-|d|0|0|1|0|0|0|1|0|0|0|0|0|0|0|0|0|
-|e|1|0|0|0|0|0|0|0|1|0|0|0|0|0|0|0|
-|f|1|0|1|0|0|0|0|0|0|1|0|0|0|0|0|0|
-|g|0|0|0|1|0|0|0|1|0|1|1|0|0|0|0|0|
-|h|0|0|0|0|0|0|1|0|0|0|0|0|0|0|1|0|
-|i|0|0|0|0|1|0|0|0|0|1|0|0|1|1|0|0|
-|j|0|0|0|0|0|1|1|0|1|0|0|0|0|0|0|0|
-|k|0|0|0|0|0|0|1|0|0|0|0|0|0|0|1|0|
-|l|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|
-|m|0|0|0|0|0|0|0|0|1|0|0|0|0|0|0|0|
-|n|0|0|0|0|0|0|0|0|1|0|0|0|0|0|1|0|
-|o|0|0|0|0|0|0|0|0|0|0|1|0|0|1|0|1|
-|p|0|0|0|0|0|0|0|0|0|0|0|1|0|0|1|0|
------------------------------------
-```
-
-Good:
-
-* constant time lookup to see if two vertexes are connected `O(1)`
-
-Bad:
-
-* space inefficient `O(n^2)`
-
-
-An adjacency matrix might be a better choice when space is less important
-than fast lookups. An adjacency list may be a better choice if space is
-a higher priority concern than time.
-
-# Traverse Every Edge
-
-To traverse every edge in both directions we can use an adjacency matrix
-and iterate through every cell in the matrix. If the cell contains a 1 to
-indicate an edge than we know that we can traverse from the edge at
-that row and column. Both directions will be represented in different cells
-in the matrix.
-
-When we visit each cell in the matrix we can flip the 1 to a 0 to ensure that
-we do not revisit a visited edge.
-
-1. Start at any vertex
-1. Iterate through list of edges.
-1. If the vertex on the other end of the edge has not been visited yet then visit it and loop until all edges are exhausted for the vertex.
-1. Remove the edge from the matrix when visiting a node
-1. Backtrack to previous vertex, and remove the edge.
-1. Visit any edge where you can backtrack safely.
-
-An example of this algorithm can be found in `./matrix.c` with accompanying tests in `./matrix_test.c`.
-
-The graph to traverse is:
-
-```plaintext
-(a)---(b)---(c)---(d)
- | \       /     /
- |  \     /     /
-(e)  \(f)/  (g)/--(h)
- |     |   / |    /
- |     |  /  |   /
-(i)---(j)/  (k) / (l)
- | \         | /   |
- |  \        |/    |
-(m)  \(n)---(o)---(p)
-```
-
-We can build a matrix that will look like the following:
-
-```bash
-| |a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|
-|a|0|1|0|0|1|1|0|0|0|0|0|0|0|0|0|0|
-|b|1|0|1|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|c|0|1|0|1|0|1|0|0|0|0|0|0|0|0|0|0|
-|d|0|0|1|0|0|0|1|0|0|0|0|0|0|0|0|0|
-|e|1|0|0|0|0|0|0|0|1|0|0|0|0|0|0|0|
-|f|1|0|1|0|0|0|0|0|0|1|0|0|0|0|0|0|
-|g|0|0|0|1|0|0|0|1|0|1|1|0|0|0|0|0|
-|h|0|0|0|0|0|0|1|0|0|0|0|0|0|0|1|0|
-|i|0|0|0|0|1|0|0|0|0|1|0|0|1|1|0|0|
-|j|0|0|0|0|0|1|1|0|1|0|0|0|0|0|0|0|
-|k|0|0|0|0|0|0|1|0|0|0|0|0|0|0|1|0|
-|l|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|1|
-|m|0|0|0|0|0|0|0|0|1|0|0|0|0|0|0|0|
-|n|0|0|0|0|0|0|0|0|1|0|0|0|0|0|1|0|
-|o|0|0|0|0|0|0|0|0|0|0|1|0|0|1|0|1|
-|p|0|0|0|0|0|0|0|0|0|0|0|1|0|0|1|0|
-```
-
-The order of traversal will be:
-
-```plaintext
-->(a)->(b)->(c)->(d)->(g)->(h)->(o)->(k)->(g)->(j)->(f)->(a)->(e)->(i)->(m)-
-                                                                           |
-|---------------------------------------------------------------------------
-->(i)->(n)->(o)->(p)->(l)->(p)->(o)->(n)->(i)->(j)->(i)->(e)->(a)->(f)->(c)-
-                                                                           |
-|---------------------------------------------------------------------------
-->(f)->(j)->(g)->(k)->(o)->(h)->(g)->(d)->(c)->(b)->(a)
-```
-
-After the traversal the matrix will have zero edges.
-
-```bash
-| |a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|
-|a|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|b|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|c|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|d|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|e|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|f|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|g|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|h|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|i|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|j|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|k|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|l|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|m|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|n|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|o|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-|p|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|0|
-```