# Graphs

# Internal

# Overview

Graphs are fundamental data structures in computer science. They map directly to a large number of problems that involve physical networks - such as the phone network or the internet, or logical networks about parallel relationships between objects in general - the order in which to execute interdependent tasks, or the analysis of social networks.

Graphs are backed by mathematical formalism. The Graph Concepts page provides a number of terms, concepts, notations and some mathematical tools that are useful when dealing with graphs. Graph Representation in Memory describes ways to represent graph nodes and edges in such a way that they can be efficiently manipulated by algorithms. The most common arrangements - adjacency lists and adjacency matrices - are discussed.

The most obvious problem that arises when dealing with graphs is to walk them. It includes searching the graph or finding paths through them, or more generically, exploring a graph to infer knowledge about it. The classical algorithms for graph exploration are breadth-first search (BFS) and depth-first search (DFS). Both these algorithms are very efficient, they are capable of exploring the graph in linear time of the number of vertices and edges O(n + m). They are described and discussed in the Graph Search page.

Building upon the basic graph search algorithms, we discuss several graph problems: computing the shortest path between two vertices using breadth-first search and then with Dijkstra's algorithm, vertex clustering heuristics involving finding connected components in an undirected graph with breadth-first search or finding strongly connected components in a directed graph with depth-first search, topological sort of a directed acyclic graph with depth-first search.

Graph cuts refer to graph partition into vertex subsets. The minimum cut problem is representative for this class of problems.

# Graph Concepts

# Graph Representation in Memory

# Graph Algorithms

- Graph Search
- Shortest Path in a Graph with breadth-first search, Dijkstra's algorithm and more
- Longest Path in a Graph
- A Path in a Graph
- Find Connected Components in an Undirected Graph with breadth-first search
- Find Strongly Connected Components in a Directed Graph with Kosaraju's two-pass algorithm, based on depth-first search
- Topological Sort of a Directed Acyclic Graph with depth-first search

- The Minimum Cut Problem
- Minimum spanning tree algorithms: Prim's Algorithm, Kruskal's Algorithm