Graph Data Structure And Algorithms

  • Last Updated : 04 Sep, 2018

GeeksforGeeks - Summer Carnival Banner

A Graph is a non-linear data structure consisting of nodes and edges. The nodes are sometimes also referred to as vertices and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph can be defined as,

A Graph consists of a finite set of vertices(or nodes) and set of Edges which connect a pair of nodes.

In the above Graph, the set of vertices V = {0,1,2,3,4} and the set of edges E = {01, 12, 23, 34, 04, 14, 13}.

Graphs are used to solve many real-life problems. Graphs are used to represent networks. The networks may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like linkedIn, Facebook. For example, in Facebook, each person is represented with a vertex(or node). Each node is a structure and contains information like person id, name, gender, locale etc.

Topics:



Introduction, DFS and BFS :
  1. Graph and its representations
  2. Breadth First Traversal for a Graph
  3. Depth First Traversal for a Graph
  4. Applications of Depth First Search
  5. Applications of Breadth First Traversal
  6. Graph representations using set and hash
  7. Find Mother Vertex in a Graph
  8. Transitive Closure of a Graph using DFS
  9. Find K cores of an undirected Graph
  10. Iterative Depth First Search
  11. Count the number of nodes at given level in a tree using BFS
  12. Count all possible paths between two vertices
  13. Minimum initial vertices to traverse whole matrix with given conditions
  14. Shortest path to reach one prime to other by changing single digit at a time
  15. Water Jug problem using BFS
  16. Count number of trees in a forest
  17. BFS using vectors & queue as per the algorithm of CLRS
  18. Level of Each node in a Tree from source node
  19. Construct binary palindrome by repeated appending and trimming
  20. Transpose graph
  21. Path in a Rectangle with Circles
  22. Height of a generic tree from parent array
  23. BFS using STL for competitive coding
  24. DFS for a n-ary tree (acyclic graph) represented as adjacency list
  25. Maximum number of edges to be added to a tree so that it stays a Bipartite graph
  26. A Peterson Graph Problem
  27. Implementation of Graph in JavaScript
  28. Print all paths from a given source to a destination using BFS
  29. Minimum number of edges between two vertices of a Graph
  30. Count nodes within K-distance from all nodes in a set
  31. Bidirectional Search
  32. Minimum edge reversals to make a root
  33. BFS for Disconnected Graph
  34. Move weighting scale alternate under given constraints
  35. Best First Search (Informed Search)
  36. Number of pair of positions in matrix which are not accessible
  37. Maximum product of two non-intersecting paths in a tree
  38. Delete Edge to minimize subtree sum difference
  39. Find the minimum number of moves needed to move from one cell of matrix to another
  40. Minimum steps to reach target by a Knight | Set 1
  41. Minimum number of operation required to convert number x into y
  42. Minimum steps to reach end of array under constraints
  43. Find the smallest binary digit multiple of given number
  44. Roots of a tree which give minimum height
  45. Stepping Numbers
  46. Clone an Undirected Graph
  47. Sum of the minimum elements in all connected components of an undirected graph
  48. Check if two nodes are on same path in a tree
  49. A matrix probability question
  50. Find length of the largest region in Boolean Matrix
  51. Iterative Deepening Search(IDS) or Iterative Deepening Depth First Search(IDDFS)

Graph Cycle :

  1. Detect Cycle in a Directed Graph
  2. Detect cycle in an undirected graph
  3. Detect cycle in a direct graph using colors
  4. Assign directions to edges so that the directed graph remains acyclic
  5. Detect a negative cycle in a Graph | (Bellman Ford)
  6. Cycles of length n in an undirected and connected graph
  7. Detecting negative cycle using Floyd Warshall
  8. Check if there is a cycle with odd weight sum in an undirected graph
  9. Check if a graphs has a cycle of odd length
  10. Clone a Directed Acyclic Graph
  11. Check loop in array according to given constraints
  12. Disjoint Set (Or Union-Find) | Set 1
  13. Union-Find Algorithm | Set 2
  14. Union-Find Algorithm | (Union By Rank and Find by Optimized Path Compression)
  15. Magical Indices in an array

Topological Sorting :

  1. Topological Sorting
  2. All topological sorts of a Directed Acyclic Graph
  3. Kahn’s Algorithm for Topological Sorting
  4. Maximum edges that can be added to DAG so that is remains DAG
  5. Longest path between any pair of vertices
  6. Longest Path in a Directed Acyclic Graph
  7. Longest Path in a Directed Acyclic Graph | Set 2
  8. Topological Sort of a graph using departure time of vertex
  9. Given a sorted dictionary of an alien language, find order of characters

Minimum Spanning Tree :

  1. Prim’s Minimum Spanning Tree (MST))
  2. Applications of Minimum Spanning Tree Problem
  3. Prim’s MST for Adjacency List Representation
  4. Kruskal’s Minimum Spanning Tree Algorithm
  5. Boruvka’s algorithm for Minimum Spanning Tree
  6. Minimum cost to connect all cities
  7. Steiner Tree
  8. Reverse Delete Algorithm for Minimum Spanning Tree
  9. Total number of Spanning Trees in a Graph
  10. Minimum Product Spanning Tree

BackTracking :

  1. Find if there is a path of more than k length from a source
  2. Tug of War
  3. The Knight-Tour Problem
  4. Rat in a Maze
  5. n-Queen’s Problem
  6. m Coloring Problem
  7. Hamiltonian Cycle
  8. Permutation of numbers such that sum of two consecutive numbers is a perfect square

Shortest Paths :

  1. Dijkstra’s shortest path algorithm
  2. Dijkstra’s Algorithm for Adjacency List Representation
  3. Bellman–Ford Algorithm
  4. Floyd Warshall Algorithm
  5. Johnson’s algorithm for All-pairs shortest paths
  6. Shortest Path in Directed Acyclic Graph
  7. Shortest path with exactly k edges in a directed and weighted graph
  8. Dial’s Algorithm
  9. Printing paths in Dijsktra’s Algorithm
  10. Shortest path of a weighted graph where weight is 1 or 2
  11. Multistage Graph (Shortest Path)
  12. Shortest path in an unweighted graph
  13. Minimize the number of weakly connected nodes
  14. Betweenness Centrality (Centrality Measure)
  15. Comparison of Dijkstra’s and Floyd–Warshall algorithms
  16. Karp’s minimum mean (or average) weight cycle algorithm
  17. 0-1 BFS (Shortest Path in a Binary Weight Graph)
  18. Find minimum weight cycle in an undirected graph
  19. Minimum Cost Path with Left, Right, Bottom and Up moves allowed
  20. Minimum edges to reverse to make path from a src to a dest
  21. Find Shortest distance from a guard in a Bank

Connectivity :



  1. Find if there is a path between two vertices in a directed graph
  2. Connectivity in a directed graph
  3. Articulation Points (or Cut Vertices) in a Graph
  4. Biconnected Components
  5. Biconnected graph
  6. Bridges in a graph
  7. Eulerian path and circuit
  8. Fleury’s Algorithm for printing Eulerian Path or Circuit
  9. Strongly Connected Components
  10. Transitive closure of a graph
  11. Find the number of islands
  12. Find the number of Islands | Set 2 (Using Disjoint Set)
  13. Count all possible walks from a source to a destination with exactly k edges
  14. Euler Circuit in a Directed Graph
  15. Count the number of non-reachable nodes
  16. Find the Degree of a Particular vertex in a Graph
  17. Check if a given graph is tree or not
  18. Minimum edges required to add to make Euler Circuit
  19. Eulerian Path in undirected graph
  20. Find if there is a path of more than k length
  21. Length of shortest chain to reach the target word
  22. Print all paths from a given source to destination
  23. Find minimum cost to reach destination using train
  24. Find if an array of strings can be chained to form a circle | Set 1
  25. Find if an array of strings can be chained to form a circle | Set 2
  26. Tarjan’s Algorithm to find strongly connected Components
  27. Number of loops of size k starting from a specific node
  28. Paths to travel each nodes using each edge (Seven Bridges of Königsberg)
  29. Number of cyclic elements in an array where we can jump according to value
  30. Number of groups formed in a graph of friends
  31. Minimum cost to connect weighted nodes represented as array
  32. Count single node isolated sub-graphs in a disconnected graph
  33. Calculate number of nodes between two vertices in an acyclic Graph by Disjoint Union method
  34. Dynamic Connectivity | Set 1 (Incremental)
  35. Check if a graph is strongly connected | Set 1 (Kosaraju using DFS)
  36. Check if a given directed graph is strongly connected | Set 2 (Kosaraju using BFS)
  37. Check if removing a given edge disconnects a graph
  38. Find all reachable nodes from every node present in a given set
  39. Connected Components in an undirected graph
  40. k’th heaviest adjacent node in a graph where each vertex has weight

Maximum Flow :

  1. Ford-Fulkerson Algorithm for Maximum Flow Problem
  2. Find maximum number of edge disjoint paths between two vertices
  3. Find minimum s-t cut in a flow network
  4. Maximum Bipartite Matching
  5. Channel Assignment Problem
  6. Push Relabel- Set 1-Introduction
  7. Push Relabel- Set 2- Implementation
  8. Karger’s Algorithm- Set 1- Introduction and Implementation
  9. Karger’s Algorithm- Set 2 – Analysis and Applications
  10. Dinic’s algorithm for Maximum Flow
  11. Max Flow Problem Introduction

STL Implementation of Algorithms :

  1. Kruskal’s Minimum Spanning Tree using STL in C++
  2. Prim’s Algorithm using Priority Queue STL
  3. Dijkstra’s Shortest Path Algorithm using STL
  4. Dijkstra’s Shortest Path Algorithm using set in STL
  5. Graph implementation using STL for competitive programming | Set 2 (Weighted graph)

Hard Problems :

  1. Graph Coloring (Introduction and Applications)
  2. Greedy Algorithm for Graph Coloring
  3. Traveling Salesman Problem (TSP) Implementation
  4. Travelling Salesman Problem (Naive and Dynamic Programming)
  5. Travelling Salesman Problem (Approximate using MST)
  6. Vertex Cover Problem | Set 1 (Introduction and Approximate Algorithm)
  7. K Centers Problem | Set 1 (Greedy Approximate Algorithm)
  8. Erdos Renyl Model (for generating Random Graphs)
  9. Clustering Coefficient in Graph Theory
  10. Chinese Postman or Route Inspection | Set 1 (introduction)
  11. Hierholzer’s Algorithm for directed graph

Misc :

  1. Number of triangles in an undirected Graph
  2. Number of triangles in directed and undirected Graph
  3. Check whether a given graph is Bipartite or not
  4. Snake and Ladder Problem
  5. Minimize Cash Flow among a given set of friends who have borrowed money from each other
  6. Boggle (Find all possible words in a board of characters)
  7. Hopcroft Karp Algorithm for Maximum Matching-Introduction
  8. Hopcroft Karp Algorithm for Maximum Matching-Implementation
  9. Minimum Time to rot all oranges
  10. Find same contents in a list of contacts
  11. Hypercube Graph
  12. Check for star graph
  13. Optimal read list for a given number of days
  14. Print all jumping numbers smaller than or equal to a given value
  15. Fibonacci Cube Graph
  16. Barabasi Albert Graph (for Scale Free Models)
  17. Construct a graph from given degrees of all vertices
  18. Degree Centrality (Centrality Measure)
  19. Katz Centrality (Centrality Measure)
  20. Mathematics | Graph theory practice questions
  21. 2-Satisfiability (2-SAT) Problem
  22. Determine whether a universal sink exists in a directed graph
  23. Number of sink nodes in a graph
  24. Largest subset of Graph vertices with edges of 2 or more colors
  25. NetworkX : Python software package for study of complex networks
  26. Generate a graph using Dictionary in Python
  27. Count number of edges in an undirected graph
  28. Two Clique Problem (Check if Graph can be divided in two Cliques)
  29. Check whether given degrees of vertices represent a Graph or Tree
  30. Finding minimum vertex cover size of a graph using binary search
  31. Stable Marriage Problem
  32. Sum of dependencies in a graph

Quick Links :

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.




My Personal Notes arrow_drop_up


Writing code in comment? Please use ide.geeksforgeeks.org, generate link and share the link here.