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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0
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1answer
10 views

Reweight general weighted graph to distinct graph for using Borůvka's

Is it possible to re-weight a generally-weighted graph to a distinctly-weighted graph to apply Borůvka's algorithm (wiki) for minimum spanning tree to it? I can't seem to think of a way to make a ...
3
votes
0answers
13 views

Birkhoff-von Neumann theorem for bistochastic digraphs

A weighted digraph (with loops) is bistochastic, iff the weights are non-negative, for all non-sink nodes, the sum of the edge weights of the out-edges is $1$, and for all non-source nodes, the sum ...
1
vote
1answer
112 views

Algorithm to find shortest path between two nodes

I want an algorithm similar to Dijkstra or Bellman-Ford for finding the shortest path between two nodes in a directed graph, but with an additional constraint. The additional constraint is that ...
1
vote
1answer
60 views

How can I evaluate an algorithm for a NP-Hard problem?

I have written a program to calculate the number of stable partition in a graph. ( That is: find which partition of the nodes does not have edges between nodes of the same block. ) The professor, ...
0
votes
1answer
18 views

Articulation vertex in complementary graph

I need to implement an algoritm, but I can't understand the theory behind it. How can I prove that if v is an articulation vertex in a graph G , that it will not be an articulation vertex in G' ...
2
votes
0answers
31 views

Constructing orthogonal latin square Parker/Knuth method

I'm working through Knuth; The Art of Computer Programming, Vol. 4 Fascicle 0 and I'm having a little trouble making sense of the method Knuth describes for computing an orthogonal latin square. The ...
0
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0answers
29 views

Finding and Grouping like children

I'm working on a dependency managemen solution for a JavaScript. I'm trying to find the best pattern for grouping similar items in a graph into their own 'modules'. Given a dependency tree like ...
2
votes
0answers
47 views

What is a best known algorithm for finding diameter of undirected graph?

What is best known algorithm (approximate or exact) for finding diameter of a large undirected graph? The diameter is defined as longest of shortest paths between any two nodes. I know that naive ...
2
votes
1answer
33 views

How to correctly contract an edge in a network?

Assuming an adjacency list as data structure of a directed Graph, it is not completely clear to me how "contraction of an edge" is defined. Let me cite the definition I do have at hand: Source: ...
1
vote
1answer
45 views

Anagrams solver based on transitions probability

I have an English dictionary (text file) and the frequency of 2-grams, 3-grams and 4-grams as the beginning of each word. I need to write an algorithm that, with a given word, calculates the possible ...
1
vote
2answers
299 views

Proving that a certain graph contains a 4-cycle

Show that if $G$ is a graph with $|E| \geq 2|V|^{3/2}$, then $G$ must contain a $4$-cycle Can someone explain/point me in the right direction on this? Let's say we say $|V|= 4$, using the ...
2
votes
1answer
23 views

Transform unstructured flow charts into structured ones

Has anyone studied the problem of converting a generic flowchart to a semantically equivalent "structured flowchart" (i.e. one that only uses the 'if' and 'while' block structure)? I can see this ...
1
vote
0answers
53 views

Maximum flow problem with non-zero lower bound

Given $G = (V,E )$ a directed graph, if $ X \subseteq V $ we write $$\begin{align*} \delta ^{+}(X) &= \{ xy\in E \mid x \in X, y\in V - X \} \\ \delta ^{-}(X) &= \delta ...
1
vote
0answers
46 views

Borůvka cleanup in linear time?

Given boruvka's algorithm: ...
3
votes
1answer
64 views

Efficient algorithm to find vertex with paths to every other vertex

$G=<V,E>$ is a directed graph. I need to write an efficient algorithm that finds a $v \in V$ such that there exists a path $\forall w \in V$ $v \rightarrow w$ ($v$ has a path to every other ...
1
vote
0answers
26 views

Does Edmonds Karp take back-edges into account?

I amb doubting with the implementation of the Edmonds-Karp implementation of the Ford-Fulkerson algorithm. This is a problem with flow networks. As I understand the algorithm, it consists on taking ...
3
votes
1answer
19 views

Do $s$-$t$ cuts partition contingent vertices?

The definition of an $s$-$t$ cut is a partition of the set of vertices $V$ into $2$ sets $(A, B)$ with $s$ in $A$ and $t$ in $B$. My understanding of set partitions is that the positioning of elements ...
1
vote
1answer
66 views

Shortest walk that covers $k$ nodes

I have the following problem, I would like to know an efficient algorithm to solve it. Suppose I have a weighted graph $G$ and a set of vertices $K$, I want to find a walk which starts at a vertex ...
0
votes
1answer
53 views

kth nearest vertex in a unweighted graph

Given an unweighted undirected graph $G$ with $10^5$ vertices and a subset $S$ of special vertices and an integer $k$, I want to find the $k$th nearest special vertex for each vertex. What algorithm ...
1
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0answers
50 views

Enumerate subtrees of a given size in a graph

Given a graph $G$ with $n$ nodes, is there an algorithm to find $m$ subtrees, each with $\lfloor n/m\rfloor$ or $\lceil n/m\rceil$ nodes, such that every node of $G$ is in exactly one tree? Other ...
0
votes
0answers
29 views

Find a path in a complete graph with cost limit and max reward

I'm looking for an algorithm to solve this problem. I have a region in which there are several areas identify by its id and x,y,z position. I've made a graph in which each vertex identifies one ot ...
2
votes
1answer
53 views

The number of maximal independent sets

An independent set is a set of vertices in a graph, no two of which are adjacent. A maximal independent set is an independent set that you can not add any vertex. I want to know if the number of all ...
0
votes
1answer
19 views

Modified Bellman Ford to find minmum cost cycle in O(E²V) time?

I'm thinking about how you can modify Bellman Ford a bit to calculate the minimum weight cycle in an undirected graph with positive weights. Note that the constraint is that the algorithm must run in ...
1
vote
1answer
75 views

Are all adjacency matrices represented by 0 and 1s? [closed]

Are there any cases when adjacency matrix should have entries other than 0 and 1?
0
votes
1answer
24 views

Unclear about proof for unique MST given graph G with distinct weights

http://homepages.math.uic.edu/~leon/cs-mcs401-s08/handouts/mst.pdf I have some trouble understanding the proof above. I understand that we assuming two MSTs, T and T', and an edge e that is the ...
5
votes
3answers
191 views

How to choose the maximum number of nodes (with constraints) from a graph

Consider a connected undirected acyclic graph $G$ with $n$ nodes and $n-1$ edges. The nodes have non-negative integer weights less than $n$. A positive integer $x$ is given and you want to choose at ...
0
votes
1answer
74 views

Dijkstra single-source shortest path $\Omega(n\log n)$?

If I have a directed graph with $n$ weighted edges, is it possible to prove that Dijkstra's single-source shortest path algorithm takes $\Omega(n\log n)$ in the worst case? I know heaps reduce ...
0
votes
2answers
27 views

Bellman–Ford negative path meaning

In the Bellman–Ford algorithm, what is the practical meaning of having a negative path between routers? I have tried searching the net but didn't find any data thanks Eli
0
votes
0answers
34 views

Evaluation in community detection

I took some evaluation metric as list in lab41.github.io/Circulo/ , and I have some doubts understanding them. as listed in the link above : Conductance, Internal density,expansion, ...
0
votes
1answer
41 views

Tournament graph

I have to prove the following assertion: given a tournament graph with $n$ vertices, $n\geq 5$, there can be made an arrangement of the arcs such that between any two vertices exists at least one way ...
2
votes
2answers
40 views

Difference between edges in Depth First Trees

I have a directed graph, where each node has an alphabetical value. The graph is to be traversed with topological DFS by descending alphabetical values (Z-A). The result is $M,N,P,O,Q,S,R,T$ (after ...
0
votes
0answers
24 views

Algorithms for verifying and solving three-coloring [duplicate]

I found the following problem that I am trying to answer: Consider the three color problem where V, vertex set of a bipartite graph. can be partitioned into three subsets such that there is no ...
1
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0answers
30 views

Getting ALL negative weight cycles of a graph using Bellman-Ford

I'm doing a min cost assignation problem to assign doctors to their working days for a hospital. After correctly getting the max flow with Ford-Fulkerson algorithm, I would like to use the cycle ...
5
votes
2answers
115 views

Is Dijkstras algorithm used in modern route-finding systems?

Is Dijkstra's algorithm used in modern route-finding systems such as Google maps or the satnav in your car? If not, then what is?
0
votes
1answer
35 views

All but Five Three Colorable

An NP Problem Named All But Five Three Colorable(AB53C) is defined as follows :- Input : Connected Graph G(V,E) The Connected Graph is AB53C, iff the Given Graph is 3-Colorable by leaving UPTO 5 ...
1
vote
1answer
56 views

Questions on Topological Sorting

Currently learning about topological sorting. My teacher gave us this problem. The answer given to us is : B,A,C,E,D,G,F,H in lexicographical order. Why does the order go from B,A,C THEN go to E ...
-1
votes
1answer
25 views

How to find a minimum cut of a network flow?

I am currently reading the lecture slides from Princeton regarding network flows but I cannot understand how they manage to find out minimum cuts from a directed graph. Could someone explain how ...
3
votes
1answer
82 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
1
vote
0answers
45 views

Set of points partitioned into max subsets of size N with no intersecting edges

Question Given a set of X kd (k-dimensional) points, find the maximum number of closed subsets of these points such that no subsets (each forming a convex hull) overlap or intersect, that each subset ...
0
votes
1answer
40 views

Vertex Cover of size k in a tree?

What is a polynomial time algorithm for finding a vertex cover of size $k$ in a tree? Would depth first or breadth first search be efficient or is there some other algorithm that finds the vertex ...
3
votes
2answers
48 views

Find a MST such that it's mostly red (original graph's edges are colored red and blue)

Consider the following problem: Given a simple, strongly-connected, weighted graph G=(V,E), of which every edge is colored either red or blue (in addition to having a numeric weight). Find an ...
7
votes
0answers
57 views

Compute a max-flow from a min-cut

We know that computing a maximum flow resp. a minimum cut of a network with capacities is equivalent; cf. the max-flow min-cut theorem. We have (more or less efficient) algorithms for computing ...
1
vote
1answer
50 views

find the shortest path between two nodes where the number of edges is minimal [closed]

Say you are given an undirected unweighted graph, where s and t are nodes from the graph. d(s,t) means the distance between s and t which outputs the number of edges. How do I find the the maximum ...
5
votes
1answer
70 views

Finding $k$ claws ($K_{1,3}$ bipartite graphs) in a graph?

Usually questions deal with claw-free graphs, but suppose we are given a graph $G$ and there are $k$ vertex-disjoing claws in the graph, how can we derive a randomised algorithm using color coding to ...
5
votes
0answers
36 views

Are there Some Pairs Shortest Paths Algorithms?

I know that there are All Pairs Shortest Paths algorithms. But I am not sure if they are effective if I am trying to solve the Pairs-Shortest-Path problem for a subset of my vertexes. The properties ...
2
votes
1answer
85 views

Find a subgraph whose edge weights sum to at least the number of nodes

Given a graph G = (V,E) every edge is assigned a real number Xe $\in$ [0,1] The sum of x variables for all edges is equal to the number of edges -1 : $\sum x_V = |V|-1$ For a subset S ...
0
votes
2answers
43 views

Why is the complexity of the BFS O(V + E)?

Consider the algorithm (from another question): ...
7
votes
2answers
281 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 $N_2$, ...
1
vote
1answer
61 views

Formulate the Marriage Problem into a Maximum-flow problem (Graph theory)

Suppose I have $M=\{1,\ldots, n\}$ men and $W = \{1, \ldots, n\}$ women and $B =\{1, \ldots, m\}$ brokers, such that each broker knows a subset of $M \times W$ and for each pair in this subset a ...
0
votes
0answers
17 views

Symbolic formalization of a rooted directed subgraph

In a directed graph $G(V, E)$; how can I define in symbols the set of subgraphs $G'=(V',E')$ of $G$ such that $G'$ has a root vertex, i.e., a vertex that sends a directed path in $G'$ to every $v\in ...