Try it on GfG Practice Given an undirected Graph, The task is to find the Bridges in this Graph. An edge in an undirected connected graph is a bridge if removing it disconnects the graph. For a disconnected undirected graph, the definition is similar, a bridge is an edge removal that increases the…

Given a list of tickets, find the itinerary in order using the given list. Note: It may be assumed that the input list of tickets is not cyclic and there is one ticket from every city except the final destination. Examples: Input: "Chennai" -> "Bangalore" "Bombay" -> "Delhi" "Goa" -> "Chennai"…

A biconnected component is a maximal biconnected subgraph. Biconnected Graph is already discussed here. In this article, we will see how to find biconnected component in a graph using algorithm by John Hopcroft and Robert Tarjan. In above graph, following are the biconnected components: 4--2 3--4…

Given a Directed Acyclic Graph (DAG), find Topological Sort of the graph. Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Topological Sorting for a graph is not possible if the graph…

Try it on GfG Practice Given a Weighted Directed Acyclic Graph (DAG) and a source vertex s in it, find the longest distances from s to all other vertices in the given graph. The longest path problem for a general graph is not as easy as the shortest path problem because the longest path problem…

Given a Directed Acyclic Graph (DAG) with V vertices and a list of directed edges represented by a 2D array edgeList[][], where each edge is defined as (u, v) meaning there is a directed edge from u to v. The task is to add the maximum number of edges to the given DAG such that it remains a DAG…

Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Topological Sorting for a graph is not possible if the graph is not a DAG. Given a DAG, print all topological sorts of the graph. For…

Given a Directed Acyclic Graph having V vertices and E edges, your task is to find any Topological Sorted order of the graph. Topological Sorted order: It is a linear ordering of vertices such that for every directed edge u -> v, where vertex u comes before v in the ordering. Example: Input: V=6 , E…

Try it on GfG Practice Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge u-v, vertex u comes before v in the ordering. Note: Topological Sorting for a graph is not possible if the graph is not a DAG. Example: Input: V = 6, edges =…

Dynamic connectivity is a data structure that dynamically maintains the information about the connected components of graph. In simple words suppose there is a graph G(V, E) in which number of vertices V is constant but number of edges E is variable. There are three ways in which we can change the…

Given n nodes and m edges (bridges) represented as a square adjacency matrix adj[][] of size n x n, where a value 1 indicates a bridge between two nodes. The task is to determine a path that uses each edge exactly once, visiting nodes accordingly. If such a path exists, print the path; otherwise,…

Try it on GfG Practice Given an array of strings, the task is to find if the given strings can be chained to form a circle. A string X can be put before another string Y in circle if the last character of X is same as first character of Y. Examples: Input: arr[] = ["geek", "king"] Output: Yes, the…

Try it on GfG Practice Given a directed graph and two vertices src and dest, the task is to count all possible walks from vertex src to vertex dest that consist of exactly k edges. The graph is represented using an adjacency matrix adj[][], where adj[i][j] = 1 indicates the presence of a directed…

Given an array of strings arr[], and two different strings start and target, representing two words. The task is to find the length of the smallest chain from string start to target, such that only one character of the adjacent words differs and each word exists in arr[]. Note: Print 0 if it is not…

Try it on GfG Practice Eulerian Path is a path in graph that visits every edge exactly once. Eulerian Circuit is an Eulerian Path which starts and ends on the same vertex. A graph is said to be eulerian if it has a eulerian cycle. We have discussed eulerian circuit for an undirected graph. In this…

Given an undirected connected graph with v nodes, and e edges, with adjacency list adj. The task is to print an Eulerian trail or circuit using Fleury's Algorithm A graph is said to be Eulerian if it contains an Eulerian Cycle, a cycle that visits every edge exactly once and starts and ends at the…

A Clique is a subgraph of graph such that all vertices in subgraph are completely connected with each other. Given a Graph, find if it can be divided into two Cliques. Example: Input : G[][] = [[0, 1, 1, 1, 0], [1, 0, 1, 0, 0], [1, 1, 0, 0, 0], [1, 0, 0, 0, 1], [0, 0, 0, 1, 0]] Output : Yes…

Given a Directed Acyclic Graph of n nodes (numbered from 1 to n) and m edges. The task is to find the number of sink nodes. A sink node is a node such that no edge emerges out of it. Examples: Input: n = 4, m = 2 Edges[] = {{2, 3}, {4, 3}} Output: 2 Explanation: Only node 1 and node 3 are sink…

Try it on GfG Practice Given a snake and ladder board, find the minimum number of dice throws required to reach the destination or last cell from the source or 1st cell. Basically, the player has total control over the outcome of the dice throw and wants to find out the minimum number of throws…

Determine whether a universal sink exists in a directed graph. A universal sink is a vertex which has no edge emanating from it, and all other vertices have an edge towards the sink. Input : v1 -> v2 (implies vertex 1 is connected to vertex 2) v3 -> v2 v4 -> v2 v5 -> v2 v6 -> v2 Output : Sink found…

This is a C++ program to generate a graph for a given fixed degree sequence. This algorithm generates a undirected graph for the given degree sequence.It does not include self-edge and multiple edges. Examples: Input : degrees[] = {2, 2, 1, 1} Output : (0) (1) (2) (3) (0) 0 1 1 0 (1) 1 0 0 1 (2) 1 0…

A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint. A maximum matching is a matching of maximum size (maximum number of edges). In a maximum matching, if any edge is added to it, it is no longer a matching. There can be more than one…

Try it on GfG Practice You are given a list of strings, words[], and an n × m matrix of characters, letters[][]. The objective is to locate all the words from words[] that can be constructed by consecutively linking adjacent characters in the matrix. You are allowed to move in any of the 8 possible…

Try it on GfG Practice Given n cities and distances between every pair of cities, select k cities to place warehouses (or ATMs or Cloud Server) such that the maximum distance of a city to a warehouse (or ATM or Cloud Server) is minimized. For example consider the following four cities, 0, 1, 2, and…

Given a directed Eulerian graph, the task is to print an Euler circuit. An Euler circuit is a path that traverses every edge of a graph exactly once, and the path ends on the starting vertex. Note: The given graph contains a Euler circuit. Example: Input : Directed graph Output: 0 3 4 0 2 1 0…

Try it on GfG Practice Chinese Postman Problem is a variation of Eulerian circuit problem for undirected graphs. An Euler Circuit is a closed walk that covers every edge once starting and ending position is same. Chinese Postman problem is defined for connected and undirected graph. The problem is…

In graph theory, the Erdos–Rényi model is either of two closely related models for generating random graphs. There are two closely related variants of the Erdos–Rényi (ER) random graph model. In the G(n, M) model, a graph is chosen uniformly at random from the collection of all graphs which have n…

Try it on GfG Practice A vertex cover of an undirected graph is a subset of its vertices such that for every edge (u, v) of the graph, either 'u' or 'v' is in the vertex cover. Although the name is Vertex Cover, the set covers all edges of the given graph. Given an undirected graph, the vertex cover…

Try it on GfG Practice Given a 2d matrix cost[][] of size n where cost[i][j] denotes the cost of moving from city i to city j. The task is to complete a tour from city 0 (0-based index) to all other cities such that we visit each city exactly once and then at the end come back to city 0 at minimum…

Graph coloring refers to the problem of coloring vertices of a graph in such a way that no two adjacent vertices have the same color. This is also called the vertex coloring problem. If coloring is done using at most m colors, it is called m-coloring. Chromatic Number: The minimum number of colors…

Try it on GfG Practice Given a connected undirected graph represented by adjacency list, adjList[][] with n nodes and m edges, with each node having a distinct label from 0 to n-1, and each adj[i] represents the list of vertices connected to vertex i. Create a clone of the graph, where each node in…

The following graph G is called a Petersen graph and its vertices have been numbered from 0 to 9. Some letters have also been assigned to vertices of G, as can be seen from the following picture: Let's consider a walk W in graph G, which consists of L vertices W1, W2, ..., WL. A string S of L…

Given an undirected and unweighted graph, find the smallest cut (smallest number of edges that disconnects the graph into two components). The input graph may have parallel edges. For example consider the following example, the smallest cut has 2 edges. A Simple Solution use Max-Flow based s-t cut…

Given n nodes of a forest (collection of trees), find the number of trees in the forest. Examples : Input : edges[] = {0, 1}, {0, 2}, {3, 4} Output : 2 Explanation : There are 2 trees 0 3 / \ \ 1 2 4 Approach : Apply DFS on every node. Increment count by one if every connected node is visited from…

Try it on GfG Practice Given a binary 2D matrix, find area of the largest region of 1s which are connected horizontally, vertically or diagonally. Examples: Input: M[][]= {{1, 0, 0, 0, 1, 0, 0}, {0, 1, 0, 0, 1, 1, 0}, {1, 1, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 1, 0, 0}, {1, 0, 0, 1, 0, 1, 1}} Output: 6…

Problem Statement : Given a graph that represents a flow network where every edge has a capacity. Also given two vertices source ‘s’ and sink ‘t’ in the graph, find the maximum possible flow from s to t with the following constraints : Flow on an edge doesn’t exceed the given capacity of the edge.An…

Given a graph which represents a flow network where every edge has a capacity. Also given two vertices source ‘s’ and sink ‘t’ in the graph, find the maximum possible flow from s to t with following constraints: Flow on an edge doesn’t exceed the given capacity of the edge. Incoming flow is equal to…

There are M transmitter and N receiver stations. Given a matrix that keeps track of the number of packets to be transmitted from a given transmitter to a receiver. If the (i; j)-th entry of the matrix is k, it means at that time the station i has k packets for transmission to station j. During a…

Try it on GfG Practice In a flow network, an s-t cut is a cut that requires the source 's' and the sink 't' to be in different subsets, and it consists of edges going from the source's side to the sink's side. The capacity of an s-t cut is defined by the sum of the capacity of each edge in the…

Given a directed graph and two vertices in it, source 's' and destination 't', find out the maximum number of edge disjoint paths from s to t. Two paths are said edge disjoint if they don't share any edge. There can be maximum two edge disjoint paths from source 0 to destination 7 in the above…