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graph_theory [2019/03/31 14:49]
graph_theory [2020/08/23 17:12] (current)
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 +====== Graph Theory ======
 +
 +Graph Theory was developed by Leonhard Euler to solve the 7 bridges of Konigsberg problem.
 +
 +A graph $G$ is a collection of vertices, $V$, and edges, $E$. 
 +
 +So we say $G = <V,E>$
 +
 +Here is an example of a graph:
 +
 +{{:graph.png?direct&400|}}
 +
 +===== Graph Terminology =====
 +
 +Incident - Edges that touch a vertex. Ex. $x$ is incident to $a$ and $b$. 
 +
 +Adjacent - Vertices that are connected. Ex. $a$ is adjacent to $b$; $b$ is adjacent to $a,c,e$
 +
 +Isolated - Vertices that are not connected. Ex. $d$ is isolated.
 +
 +{{:graph_term_1.png?direct&600|}}
 +
 +The are two types of graphs, undirected and directed. In a directed graph, order matters. 
 +
 +{{:graph_term_2.png?direct&600|}}
 +
 +There are two types of "walks". X-Y walks and closed walks. 
 +
 +{{::graph_term_4.png?direct&600|}}
 +
 +{{::graph_term_3.png?direct&600|}}
 +
 +{{::graph_term_5.png?direct&600|}}
 +
 +**def:** the number of edges incident to a node is called the **degree** of a node.
 +
 +**def:** A graph is **simple** if it has no loops of multiple edges.
 +
 +Graph Coloring Problem
 +
 +Given graph $G$ and $k$ colors assign a color to each node so that the adjacent nodes get different colors.
 +
 +Def. Minimum value of $k$ for which such a coloring exists is the chromatic value. $χ(G)$
 +
 +Basic graph coloring algorithm for $G=(V,E)$
 +
 +    - Order the nodes $V_1, V_2 .. V_n$
 +    - Order the colors $C_1, C_2 .. C_k$
 +    - For $i = 1,2 ... n$ assign lowest legal color for $V1$
 +
 +===== Topological Sorting =====
 +
 +[[https://www.youtube.com/watch?v=jonc-AGbPQo|Intro video on topoligical sorting.]]
 +
 +==== Kahn's Algorithm for Topological Sorting ====
 +
 +[[https://www.youtube.com/watch?v=tFpvX8T0-Pw|Video on Kahn's algorithm of topological sorting.]]
 +
 +Everyime we visit a vertex, we decrement the number of depencies it has.
 +
 +This is a good explanation of the Alien Dictionary problem: [[https://leetcode.com/problems/alien-dictionary/discuss/157298/C%2B%2B-BFS-and-Topoligical-Sort-with-explanation|link]].