Questions tagged [graphs]

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

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21
votes
2answers
10k views

Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight?

If a weighted graph $G$ has two different minimum spanning trees $T_1 = (V_1, E_1)$ and $T_2 = (V_2, E_2)$, then is it true that for any edge $e$ in $E_1$, the number of edges in $E_1$ with the same ...
45
votes
4answers
54k 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 $v'$. ...
30
votes
1answer
11k views

How hard is counting the number of simple paths between two nodes in a directed graph?

There is an easy polynomial algorithm to decide whether there is a path between two nodes in a directed graph (just do a routine graph traversal with, say, depth-first-search). However it seems that, ...
25
votes
3answers
12k 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 ...
9
votes
1answer
6k 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 $2^{nd}$-...
32
votes
8answers
48k 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 ...
28
votes
4answers
32k views

The time complexity of finding the diameter of a graph

What is the time complexity of finding the diameter of a graph $G=(V,E)$? ${O}(|V|^2)$ ${O}(|V|^2+|V| \cdot |E|)$ ${O}(|V|^2\cdot |E|)$ ${O}(|V|\cdot |E|^2)$ The diameter of a ...
45
votes
3answers
59k 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 ...
33
votes
5answers
6k 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 $n$ ...
26
votes
1answer
16k views

Is the k-clique problem NP-complete?

In this Wikipedia article about the Clique problem in graph theory it states in the beginning that the problem of finding a clique of size K, in a graph G is NP-complete: Cliques have also been ...
24
votes
3answers
25k 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 ...
12
votes
2answers
6k 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 ...
8
votes
1answer
350 views

How hard is finding the shortest path in a graph matching a given regular language?

Suppose we are given a directed graph $G = (V, E)$ with edge weights $w : E \rightarrow \mathbb{R}$ (we can assume there are no negative cycles) and edge labels $\ell : E \rightarrow \Sigma$ from some ...
17
votes
1answer
16k 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 ...
6
votes
1answer
5k 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 ...
5
votes
2answers
647 views

Finding a subset in bipartite graph violating Hall's condition

We are given a bipartite graph of $n \leq 200$ vertices in both the first and the second partite set. Let $U$ be some set of vertices in the first set, and $V$ those vertices from the second that are ...
18
votes
7answers
51k views

Why can't DFS be used to find shortest paths in unweighted graphs?

I understand that using DFS "as is" will not find a shortest path in an unweighted graph. But why is tweaking DFS to allow it to find shortest paths in unweighted graphs such a hopeless prospect? ...
6
votes
1answer
2k views

Does Ford-Fulkerson always produce the left-most min-cut

When using Ford-Fulkerson to find max-flow between s and t, the exact choice of flow-graph depends on which paths are found. However, if you then use the left-over residual graph to produce a min-cut ...
3
votes
1answer
461 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 ...
0
votes
1answer
3k views

Number of Different AVL Tree

I studying the related question. https://stackoverflow.com/questions/13500560/number-of-ways-to-create-an-avl-tree-with-n-nodes-and-l-leaf-node but it's not so general. In-fact, We want to know ...
37
votes
4answers
43k 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 ...
14
votes
2answers
6k views

Proving a binary tree has at most $\lceil n/2 \rceil$ leaves

I'm trying to prove that a binary tree with $n$ nodes has at most $\left\lceil \frac{n}{2} \right\rceil$ leaves. How would I go about doing this with induction? For people who were following in the ...
14
votes
1answer
4k 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 ...
10
votes
4answers
740 views

Recovering a point embedding from a graph with edges weighted by point distance

Suppose I give you an undirected graph with weighted edges, and tell you that each node corresponds to a point in 3d space. Whenever there's an edge between two nodes, the weight of the edge is the ...
8
votes
2answers
6k views

Updating an MST $T$ when the weight of an edge not in $T$ is decreased

Given an undirected, connected, weighted graph $G = (V,E,w)$ where $w$ is the weight function $w: E \to \mathbb{R}$ and a minimum spanning tree (MST) $T$ of $G$. Now we decrease the weight by $k$ of ...
6
votes
2answers
15k 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 ...
4
votes
1answer
170 views

Between every two MST's there's a series of “nearby” MST's

Given undirected connected graph $G=(V,E)$ and a weight function $w:E\to\mathbb{R}$, two MST's $T_1, T_2$ are nearby if there exists $e\in T, e'\in T'$ such that $T'=(T-\{e\})\cup\{e'\}$. Prove that ...
3
votes
1answer
3k views

What edges are not in any MST?

This is a homework question. I do not want the solution - I'm offering the solution I've been thinking of and wish to know whether is it good or why is it flawed. Consider a weighted undirected graph....
3
votes
1answer
2k 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 $k$ ...
15
votes
1answer
17k 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 ...
6
votes
2answers
6k views

How to find spanning tree of a graph that minimizes the maximum edge weight?

Suppose we have a graph G. How can we find a spanning tree that minimizes the maximum weight of all the edges in the tree? I am convinced that by simply finding an MST of G would suffice, but I am ...
3
votes
1answer
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) ...
1
vote
2answers
408 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 ...
19
votes
5answers
15k views

Maximum Independent Set of a Bipartite Graph

I'm trying to find the Maximum Independent Set of a Biparite Graph. I found the following in some notes "May 13, 1998 - University of Washington - CSE 521 - Applications of network flow": Problem: ...
15
votes
1answer
4k views

$O(n^{k-1}$) algorithm for K-clique problem

Clique problem is a well known $NP$-complete problem where the size of the required clique is part of the input. However, k-clique problem has a trivial polynomial time algorithm ($O(n^k)$ when $k$ is ...
9
votes
1answer
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 ...
7
votes
1answer
5k views

How to find the maximum independent set of a directed graph?

I'm trying to solve this problem. Problem: Given $n$ positive integers, your task is to select a maximum number of integers so that there are no two numbers $a, b$ in which $a$ is divisible by $b$...
4
votes
1answer
3k 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 ...
3
votes
1answer
5k 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 ...
8
votes
4answers
23k views

Every simple undirected graph with more than $(n-1)(n-2)/2$ edges is connected

If a graph with $n$ vertices has more than $\frac{(n-1)(n-2)}{2}$ edges then it is connected. I am a bit confused about this question, since I can always prove that for a graph to connected you need ...
6
votes
2answers
3k views

Example of graph with exponential many s-t minpaths and min cuts

I am trying to find a graph in which both s-t minpaths and min cuts are exponential. Individually I found examples in which s-t minpaths and s-t min cuts are exponential. Can some one provide me an ...
3
votes
2answers
270 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 ...
2
votes
2answers
2k views

CNF Generator for Factoring Problems

I've been reading these: Fast Reduction from RSA to SAT CNF Generator for Factoring Problems (Also have C code implementation) I don't understand how the reduction from FACT to $3\text{-SAT}$ works. ...
3
votes
1answer
3k views

Using coloring optimization or coloring decision to solve coloring search

How can you show that coloring search can be solved by making a polynomial number of calls to the solution for coloring optimization or coloring decision? (Coloring search is the algorithm to color ...
3
votes
1answer
397 views

Restricted version of vertex cover

I am interested in the complexity of the restricted version of the vertex cover problem below: Instance: A bipartite graph $G =(L, R, E)$ and an integer $K$. Question: Is there $S \subset L$, $|S| \...
3
votes
2answers
2k 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 ...
4
votes
3answers
2k views

Reduce Vertex Cover with size k to Vertex Cover with size n/2

Disclaimer: This is a homework question. I would like to reduce vertex cover problem to the following problem: $$L = \{G \mid G\text{ has a vertex cover of size } |V(G)|/2\}\,.$$ I have divided the ...
4
votes
1answer
99 views

First Order interpretation of arbitrary structures as a graph

I am currently trying to get some intuition on the concept of First Order reductions, and have come across this exercise question by Immerman, dubbed "Everything is a Graph". Given some arbitrary ...
4
votes
3answers
2k views

Number of ways to fill a 2xN grid with M colors

This question was asked in the onsite regionals for ACM ICPC 2013 at Amritapuri. In short, the question asked to find the number of ways to fill a $ 2\times N$ grid with $M$ colors such that no two ...
0
votes
1answer
519 views

Is this “cycle” condition sufficient for unique minimum spanning tree?

Given a connected, undirected, weighted graph $G$, the condition The maximum-weight edge in any cycle of $G$ is unique. is not necessary for $G$ to have a unique minimum spanning tree (MST). ...

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