Graphs

	A few definitions:

		Vertices -a point or a place

		Edge - connects any two vertices

		Adjacency - two vertices attached by an edge

		Path - a sequence of edges from one vertex to another

		Connected Graph - at least one path from every vertex to every other vertex

		Cyclic Graph - can finish up where you started

		Directed Graph - can only travel one direction on an edge

		Un-Directed Graph - can travel either direction on an edge
		
		Weighted Graph - each edge holds a value (cost, time, distance)
		
		UnWeighted Graph - no value for the edges

	Adjacency Matrix
	
		An adjacency matrix is created by listing all vertices in rows and columns, then marking
		the intersecting cell for any pair that are adjacent.
		
			|   | A | B | C | D | E | F | G | H | I | J |
			|---|---|---|---|---|---|---|---|---|---|---|
			| A |   | x | x | x |   |   |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| B | x |   |   |   |   | x | x |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| C | x |   |   |   | x |   |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| D | x |   |   |   |   |   |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| E |   |   | x |   |   |   |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| F |   | x |   |   |   |   | x | x | x |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| G |   | x |   |   |   |   |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| H |   |   |   |   |   | x |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| I |   |   |   |   |   | x |   |   |   |   |
			|---|---|---|---|---|---|---|---|---|---|---|
			| J |   |   |   |   |   | x |   |   |   |   |

	Adjacency List

		An adjaceny list is created by first listing all vertices in rows, then listing all vertices
		that can be directly visited from each vertex.

			A-> B, C, D
			B-> A, F, G
			C-> A, E
			D-> A
			E-> C
			F-> B, H, I, J
			G-> B
			H-> F
			I-> F
			J-> F

	Searching a graph

		Depth-First Search
		
			follow a path as far as possible, then backup to previous vertex and follow another path.
			Use Stack to keep track of which vertices still have paths to follow.

			1. visit a vertex
			2. mark it
			3. push it onto a stack
			4. visit an adjacent vertex
				if possible - loop to step 2
				if not - pop the next vertex
			5. When stack is empty, search is done

	Breadth-First Search
	
		from a vertex, visit all adjacent vertices.  Use Queue to keep track of which vertices still
		have adjacent vertices to visit.

		1. visit a vertex
		2. visit all adjacent vertices, placing them in a queue
		3. pop from the queue
			if possible - loop back to step 2
			if not - done