# Questions tagged [graphs]

Questions about graphs, discrete structures of nodes which are connected by edges. Popular flavors are trees and networks with edge capacity.

216 questions
Filter by
Sorted by
Tagged with
45k views

### Longest path in an undirected tree with only one traversal

There is this standard algorithm for finding longest path in undirected trees using two depth-first searches: Start DFS from a random vertex $v$ and find the farthest vertex from it; say it is $v'$. ...
11k views

### Retrieving the shortest path of a dynamic graph

I'm studying shortest paths in directed graphs currently. There are many efficient algorithms for finding the shortest path in a network, like dijkstra's or bellman-ford's. But what if the graph is ...
5k views

### Finding the k-shortest path between two nodes

Given a weighted digraph $G=V,E$, and a weight function, $d(u,v)$, one can normally use Dijkstra's algorithm to obtain the shortest path. What I am interested in, is how to obtain the $2^{nd}$-...
37k views

### Why does Dijkstra's algorithm fail on a negative weighted graphs? [duplicate]

I know this is probably very basic, I just can't wrap my head around it. We recently studied about Dijkstra's algorithm for finding the shortest path between two vertices on a weighted graph. My ...
13k views

### Find the Simple Cycles in a Directed Graph

This problem, for me, looks very interesting. It was about to find a simple cycle (i.e. cycle where are not repeat nodes) in a directed graph. My solution is going like this, i.e, this graph is a ...
4k views

### Find a 3-colouring using the 3-colourability decision problem

I was learning about NP problems. I read that for many problems, like Clique, we can easily convert its decision problem to derive a solution of search problem. (For Clique problem, you only need to ...
4k views

### Enumerate all non-isomorphic graphs of a certain size

I'd like to enumerate all undirected graphs of size $n$, but I only need one instance of each isomorphism class. In other words, I want to enumerate all non-isomorphic (undirected) graphs on $n$ ...
15k views

### When is the minimum spanning tree for a graph not unique

Given a weighted, undirected graph G: Which conditions must hold true so that there are multiple minimum spanning trees for G? I know that the MST is unique when all of the weights are distinct, but ...
360 views

### Expected distance between tree nodes

I have been given a tree with n nodes and n-1 edges with it's weight. There are two people A and B. I have been given a list of nodes of size k. A will pick a random node x from this list and B will ...
39k views

### Algorithm to find diameter of a tree using BFS/DFS. Why does it work?

This link provides an algorithm for finding the diameter of an undirected tree using BFS/DFS. Summarizing: Run BFS on any node s in the graph, remembering the node u discovered last. Run BFS from u ...
12k views

### Is there an algorithm to find all the shortest paths between two nodes?

Given a directed graph, Dijkstra or Bellman-Ford can tell you the shortest path between two nodes. What if there are two (or n) paths that are shortest, is there an algorithm that will tell you all ...
1k views

### Dijkstra's algorithm to compute shortest paths using k edges?

I am aware of using Bellman-Ford on a graph $G=(V,E)$ with no negative cycles to find the single-source single-destination shortest paths from source $s$ to target $t$ (both in $V$) using at most $k$ ...
16k views

### Find the longest path from root to leaf in a tree

I have a tree (in the graph theory sense), such as the following example: This is a directed tree with one starting node (the root) and many ending nodes (the leaves). Each of the edge has a length ...
3k views

### Why can't we find shortest paths with negative weights by just adding a constant so that all weights are positive?

I'm currently reading introduction to algorithms and came by Johnson’s algorithm that depends on making sure that all paths are positive. the algo depends on finding a new weight function (w') that ...
2k views

### Efficiently enumerating all paths from i to j of given length in a graph

I've been trying to efficiently solve this problem : given a integer p > 0 and a directed graph whose nodes are 0, ..., N-1, enumerate (not simply count) all the paths (not necessarily elementary) ...
211 views

### Equivalent definition of minimal spanning tree

Prove that $T$ is MST $\Leftrightarrow$ for any edge $uv \notin T$, $uv$ has the maximal weight on the cycle created by adding $uv$ to $T$. It's my attempt to prove $\Rightarrow$: Consider the cycle ...
36k views

### Algorithm that finds the number of simple paths from $s$ to $t$ in $G$

Can anyone suggest me a linear time algorithm that takes as input a directed acyclic graph $G=(V,E)$ and two vertices $s$ and $t$ and returns the number of simple paths from $s$ to $t$ in $G$. I have ...
2k views

### Are all MST minimum spanning trees reachable by Kruskal and Prim?

I believe this is true but have not been able to get a formal proof for either. But is it true that any minimum spanning tree is reachable by applying Kruskal's algorithm? Similarly, is this true for ...
1k views

### What is the most efficient algorithm and data structure for maintaining connected component information on a dynamic graph?

Say I have an undirected finite sparse graph, and need to be able to run the following queries efficiently: $IsConnected(N_1, N_2)$ - returns $T$ if there is a path between $N_1$ and $N_2$, otherwise ...
113 views

### Single-source shortest paths with even weight

I need help to find an algorithm that calculates the single-source shortest paths in a graph, with an extra condition that the weight of the path has to be even. In another words, we have to find the ...
4k views

### single algorithm to work on both directed and undirected graph to detect cycles?

I have been trying to implement an algorithm to detect cycles (probably how many of them) in a directed and undirected graph. That is the code should apply for both ...
1k views

### Hamiltonian path in grid graph

Here is my situation. I have a grid-type graph with obstacles. Every move (horizontally, vertically or diagonally with a range of 1) has a cost of exactly 1 (the graph is not weighted) provided that ...
2k views

### Algorithm for solving incremental max flow problem

I am working on a project where I need to be able to compute the maximum flow between two nodes in a graph after one of the edge weights has been incremented or decremented by 1. The graph is directed ...
239 views

### Vertex cover of bipartite graph

A vertex cover is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A minimum vertex cover is a vertex cover with minimal cardinality. From codeforces, ...
5k views

### Getting parallel items in dependency resolution

I have implemented a topological sort based on the Wikipedia article which I'm using for dependency resolution, but it returns a linear list. What kind of algorithm can I use to find the independent ...
3k views

### Where to get graphs to test my search algorithms against?

I am implementing a set of path finding algorithms such as Dijkstra's, Depth First, etc. At first I used a couple of self made graphs, but now I'd like to take the challenge a bit further and thus I'...
28k views

### Getting negative cycle using Bellman Ford

I have to find a negative cycle in a directed weighted graph. I know how the Bellman Ford algorithm works, and that it tells me if there is a reachable negative cycle. But it does not explicitly name ...
2k views

### Shortest non intersecting path for a graph embedded in a euclidean plane (2D)

What algorithm would you use to find the shortest path of a graph, which is embedded in an euclidean plane, such that the path should not contain any self-intersections (in the embedding)? For ...
8k views

### Do you get DFS if you change the queue to a stack in a BFS implementation?

Here is the standard pseudocode for breadth first search: ...
972 views

### How many shortest distances change when adding an edge to a graph?

Let $G=(V,E)$ be some complete, weighted, undirected graph. We construct a second graph $G'=(V, E')$ by adding edges one by one from $E$ to $E'$. We add $\Theta(|V|)$ edges to $G'$ in ...
1k views

### Finding shortest paths in undirected graphs with possibly negative edge weights

The book "Algorithms" by Robert Sedgewick and Kevin Wayne hinted that (see the quote below) there are efficient algorithms for finding shortest paths in undirected graphs with possibly negative edge ...
2k views

6k views

### Kosaraju’s Algorithm - why transpose? [duplicate]

In directed graph, to find strongly connected components why do we have to transpose adjacency matrix (reverses the direction of all edges) if we could use reversed list of nodes by they finishing ...
1k views

### How to approach Dynamic graph related problems

I asked this question at generic stackoverflow and I was directed here. It will be great if some one can explain how to approach partial or fully dynamic graph problems in general. For example: ...
6k views

### Converting a digraph to an undirected graph in a reversible way

I am looking for an algorithm to convert a digraph (directed graph) to an undirected graph in a reversible way, ie the digraph should be reconstructable if we are given the undirected graph. I ...
3k views

### Correctness of Strongly Connected Components algorithm for a directed graph

I have been reading up on algorithm for finding the strongly connected components in a directed graph $G=(V,E)$. It considers two DFS search and the second step is transposing the original graph $G^T$....
318 views

### Transition coverage for a DFA

Let $G$ be a directed graph, with a single source node $s$. I want to find a collection of paths that cover every edge of $G$ (i.e., every edge of $G$ appears in at least one of these paths), where ...
3k views

### Optimal algorithm to traverse all paths in the order of shortest path

I have to generate all possible paths in a directed, acyclic weighted graph with edge costs. I also have to sort them in order of shortest path. The simplest way that comes to mind is to do a depth-...
154 views

### Given all pairs shortest paths matrix, find graph with minimum total sum of edges

I was looking at some problems about graphs, and I got stuck on this one. Namely, we have given matrix of size $N \cdot N$ representing the length of the shortest path in undirected graph between some ...
1k views

### Kosaraju's Algorithm-Strongly connected components

In Kosaraju's Algorithm, using first dfs (traversing on reverse graph) we calculate finishing time of nodes, and then traverse (actual graph) in reverse order of finishing times. why not without ...
11k views

### If all edges are of equal weight, can one use BFS to obtain a minimal spanning tree?

If given that all edges in a graph $G$ are of equal weight $c$, can one use breadth-first search (BFS) in order to produce a minimal spanning tree in linear time? Intuitively this sounds correct, as ...
583 views

### Find a path from s to t using as few red nodes as possible

Was doing a little interview prep. Given an undirected graph G, such that each node is colored red or blue and |E|≥|V|, find a path in O(|E|) time such that starting and ending at 2 blue nodes, s and ...
191 views

### Why Djikstra's algorithm is said to have $\mathcal{O}(|V|^2)$ complexity?

Djikstra's algorithm assigns some number to non-removed vertex each time it finds a path from removed vertex to it. Number of assignments is $\mathcal{O}(|V|^2)$. However, complexity of assignment is ...
413 views

### Proving that finding wheel subgraphs is NP-complete

Can you help me with this problem ? Given an undirected graph $G$ and an integer $n$, prove that determining whether the graph has wheel on $n$ vertices $W_{n}$ (a wheel $W_{i}$ is such that $i$ ...
395 views

### Is it possible to reconstruct graph if we have given matrix of shortest pairs

I'm trying to reconstruct graph if we have given the result of floyd-warshall algorithm, more formally: Let's say we have given undirected weighted tree (graph without cycles) with $N$ nodes, such ...
550 views

### NP-completeness of graph isomorphism through edge contractions with an edge validity condition

Given Graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$. Can a graph isomorphic to $H$ be obtained from $G$ by a sequence of edge contractions ? We know this problem is NP-complete. What about if only a subset ...
2k views

### Tarjan's Strongly Connected Component algorithm

I am trying to understand Tarjan's strongly connected component algorithm and I have a few questions (the line numbers I am referring to are from Algoritmy.net): On line 33 why is ...
Let $G$ be an undirected connected weighted graph, and let $T$ be a minimum spanning tree of $G$ with edge weights: $w_1 \le w_2 \le ... \le w_{n-1}$. Now let $T'$ be some other spanning tree of $G$ (...