Graph Algorithms Last Updated : 21 Jun, 2025 Comments Improve Suggest changes Like Article Like Report Graph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social network, a computer network, a network of locations used in GPS and many more examples where different nodes or vertices are connected without any hierarchic or constraint on structure.BasicsGraph and its representationsBFS and DFS Breadth First TraversalDepth First Traversal Difference between BFS and DFSRotten TomatoesIslands in a GraphFlood FillCheck for BipartiteWord Ladder Snakes and LadderWater Jug problemPacific Atlantic Water FlowShortest Path in Binary MatrixClone a Graph Transitive Closure of a Graph using DFSCycles Cycle in a Directed GraphCycle in an undirected graphCycle in a graph using colorsNegative cycle in a Graph | (Bellman Ford)Cycles of length nClone a Directed Acyclic Graph Disjoint Set Data Structure or Union-Find AlgorithmShortest Path Dijkstra’s shortest path aBellman–Ford Floyd Warshall Johnson’s algorithm Shortest Path in Directed Acyclic GraphDial’s AlgorithmMultistage Graph (Shortest Path)Shortest path in an unweighted graphMinimum mean weight cycle algorithm0-1 BFS (Shortest PathMinimum weight cycle D’Esopo-Pape AlgorithmMinimum Spanning TreePrim’s Minimum Spanning Tree (MST)Kruskal’s Minimum Spanning Tree Prim’s vs Kruskal’s algorithm for MSTApplications of Minimum Spanning Tree ProblemMinimum cost to connect all citiesTotal number of Spanning Trees in a GraphMinimum Product Spanning TreeReverse Delete Algorithm for Minimum Spanning TreeBoruvka’s algorithm for Minimum Spanning TreeTopological SortingTopological SortingAll topological sorts of a Directed Acyclic GraphKahn’s Algorithm for Topological SortingMaximum edges that can be added to DAG so that is remains DAGLongest Path in a Directed Acyclic GraphTopological Sort of a graph using departure time of vertexFind Itinerary from a given list of ticketsConnectivity in GraphArticulation Points (or Cut Vertices) in a GraphBiconnected ComponentsBridges in a graphEulerian path and circuitFleury’s Algorithm for printing Eulerian Path or CircuitStrongly Connected ComponentsCount all possible walks from a source to a destination with exactly k edgesEuler Circuit in a Directed GraphLength of shortest chain to reach the target wordFind if an array of strings can be chained to form a circleTarjan’s Algorithm to find strongly connected ComponentsPaths to travel each nodes using each edge (Seven Bridges of Königsberg)Dynamic Connectivity | Set 1 (Incremental)Maximum Flow in GraphMax Flow Problem IntroductionFord-Fulkerson Algorithm for Maximum Flow ProblemFind maximum number of edge disjoint paths between two verticesFind minimum s-t cut in a flow networkMaximum Bipartite MatchingChannel Assignment ProblemIntroduction to Push Relabel AlgorithmKarger’s Algorithm- Set 1- Introduction and ImplementationDinic’s algorithm for Maximum FlowSome must do Problems Find length of the largest region in Boolean MatrixCount number of trees in a forestA Peterson Graph ProblemClone an Undirected GraphGraph Coloring (Introduction and Applications)Traveling Salesman Problem (TSP) ImplementationVertex Cover Problem | Set 1 (Introduction and Approximate Algorithm)K Centers Problem | Set 1 (Greedy Approximate Algorithm)Erdos Renyl Model (for generating Random Graphs)Chinese Postman or Route Inspection | Set 1 (introduction)Hierholzer’s Algorithm for directed graphCheck whether a given graph is Bipartite or notSnake and Ladder ProblemBoggle (Find all possible words in a board of characters)Hopcroft Karp Algorithm for Maximum Matching-IntroductionMinimum Time to rot all orangesConstruct a graph from given degrees of all verticesDetermine whether a universal sink exists in a directed graphNumber of sink nodes in a graphTwo Clique Problem (Check if Graph can be divided in two Cliques)If you are looking for difficulty-wise list of problems, please refer to Graph Data Structure.Some QuizzesQuizzes on Graph TraversalQuizzes on Graph Shortest PathQuizzes on Graph Minimum Spanning TreeQuizzes on GraphsQuick Links :Top 10 Interview Questions on Depth First Search (DFS)Some interesting shortest path questionsPractice Problems on Graphs Recommended:Learn Data Structure and Algorithms | DSA Tutorial Comment More infoAdvertise with us Next Article Introduction to Graph Data Structure H harendrakumar123 Follow Improve Article Tags : Graph DSA Practice Tags : Graph Similar Reads Graph Algorithms Graph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net 3 min read Introduction to Graph Data Structure Graph Data Structure is a non-linear data structure consisting of vertices and edges. It is useful in fields such as social network analysis, recommendation systems, and computer networks. In the field of sports data science, graph data structure can be used to analyze and understand the dynamics of 15+ min read Graph and its representations A Graph is a non-linear data structure consisting of vertices and edges. The vertices are sometimes also referred to as nodes and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph is composed of a set of vertices( V ) and a set of edges( E ). The graph is den 12 min read Types of Graphs with Examples A graph is a mathematical structure that represents relationships between objects by connecting a set of points. It is used to establish a pairwise relationship between elements in a given set. graphs are widely used in discrete mathematics, computer science, and network theory to represent relation 9 min read Basic Properties of a Graph 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. The basic properties of a graph include: Vertices (nodes): The points where edges meet in a graph are kn 4 min read Applications, Advantages and Disadvantages of Graph Graph is a non-linear data structure that contains nodes (vertices) and edges. A graph is a collection of set of vertices and edges (formed by connecting two vertices). A graph is defined as G = {V, E} where V is the set of vertices and E is the set of edges. Graphs can be used to model a wide varie 7 min read Transpose graph Transpose of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G. That is, if G contains an edge (u, v) then the converse/transpose/reverse of G contains an edge (v, u) and vice versa. Giv 9 min read Difference Between Graph and Tree Graphs and trees are two fundamental data structures used in computer science to represent relationships between objects. While they share some similarities, they also have distinct differences that make them suitable for different applications. Difference Between Graph and Tree What is Graph?A grap 2 min read BFS and DFS on GraphBreadth First Search or BFS for a GraphGiven a undirected graph represented by an adjacency list adj, where each adj[i] represents the list of vertices connected to vertex i. Perform a Breadth First Search (BFS) traversal starting from vertex 0, visiting vertices from left to right according to the adjacency list, and return a list conta 15+ min read Depth First Search or DFS for a GraphIn Depth First Search (or DFS) for a graph, we traverse all adjacent vertices one by one. When we traverse an adjacent vertex, we completely finish the traversal of all vertices reachable through that adjacent vertex. This is similar to a tree, where we first completely traverse the left subtree and 13 min read Applications, Advantages and Disadvantages of Depth First Search (DFS)Depth First Search is a widely used algorithm for traversing a graph. Here we have discussed some applications, advantages, and disadvantages of the algorithm. Applications of Depth First Search:1. Detecting cycle in a graph: A graph has a cycle if and only if we see a back edge during DFS. So we ca 4 min read Applications, Advantages and Disadvantages of Breadth First Search (BFS)We have earlier discussed Breadth First Traversal Algorithm for Graphs. Here in this article, we will see the applications, advantages, and disadvantages of the Breadth First Search. Applications of Breadth First Search: 1. Shortest Path and Minimum Spanning Tree for unweighted graph: In an unweight 4 min read Iterative Depth First Traversal of GraphGiven a directed Graph, the task is to perform Depth First Search of the given graph.Note: Start DFS from node 0, and traverse the nodes in the same order as adjacency list.Note : There can be multiple DFS traversals of a graph according to the order in which we pick adjacent vertices. Here we pick 10 min read BFS for Disconnected GraphIn the previous post, BFS only with a particular vertex is performed i.e. it is assumed that all vertices are reachable from the starting vertex. But in the case of a disconnected graph or any vertex that is unreachable from all vertex, the previous implementation will not give the desired output, s 14 min read Transitive Closure of a Graph using DFSGiven a directed graph, find out if a vertex v is reachable from another vertex u for all vertex pairs (u, v) in the given graph. Here reachable means that there is a path from vertex u to v. The reach-ability matrix is called transitive closure of a graph. For example, consider below graph: GraphTr 8 min read Difference between BFS and DFSBreadth-First Search (BFS) and Depth-First Search (DFS) are two fundamental algorithms used for traversing or searching graphs and trees. This article covers the basic difference between Breadth-First Search and Depth-First Search.Difference between BFS and DFSParametersBFSDFSStands forBFS stands fo 2 min read Cycle in a GraphDetect Cycle in a Directed GraphGiven the number of vertices V and a list of directed edges, determine whether the graph contains a cycle or not.Examples: Input: V = 4, edges[][] = [[0, 1], [0, 2], [1, 2], [2, 0], [2, 3]]Cycle: 0 → 2 → 0 Output: trueExplanation: The diagram clearly shows a cycle 0 → 2 → 0 Input: V = 4, edges[][] = 15+ min read Detect cycle in an undirected graphGiven an undirected graph, the task is to check if there is a cycle in the given graph.Examples:Input: V = 4, edges[][]= [[0, 1], [0, 2], [1, 2], [2, 3]]Undirected Graph with 4 vertices and 4 edgesOutput: trueExplanation: The diagram clearly shows a cycle 0 → 2 → 1 → 0Input: V = 4, edges[][] = [[0, 8 min read Detect Cycle in a directed graph using colorsGiven a directed graph represented by the number of vertices V and a list of directed edges, determine whether the graph contains a cycle.Your task is to implement a function that accepts V (number of vertices) and edges (an array of directed edges where each edge is a pair [u, v]), and returns true 9 min read Detect a negative cycle in a Graph | (Bellman Ford)Given a directed weighted graph, your task is to find whether the given graph contains any negative cycles that are reachable from the source vertex (e.g., node 0).Note: A negative-weight cycle is a cycle in a graph whose edges sum to a negative value.Example:Input: V = 4, edges[][] = [[0, 3, 6], [1 15+ min read Cycles of length n in an undirected and connected graphGiven an undirected and connected graph and a number n, count the total number of simple cycles of length n in the graph. A simple cycle of length n is defined as a cycle that contains exactly n vertices and n edges. Note that for an undirected graph, each cycle should only be counted once, regardle 10 min read Detecting negative cycle using Floyd WarshallWe are given a directed graph. We need compute whether the graph has negative cycle or not. A negative cycle is one in which the overall sum of the cycle comes negative. Negative weights are found in various applications of graphs. For example, instead of paying cost for a path, we may get some adva 12 min read Clone a Directed Acyclic GraphA directed acyclic graph (DAG) is a graph which doesn't contain a cycle and has directed edges. We are given a DAG, we need to clone it, i.e., create another graph that has copy of its vertices and edges connecting them. Examples: Input : 0 - - - > 1 - - - -> 4 | / \ ^ | / \ | | / \ | | / \ | 12 min read Like