Questions about properties of and problems on graphs, discrete data structures that have the form of nodes connected by edges, that is networks.

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8 views

Maximum bipartite matching when some nodes must be matched

Consider the problem of finding a maximum cardinality bipartite matching under the additional condition that some set $S$ of nodes (all lying on the same side of the bipartition) must be matched. ...
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1answer
31 views

What does pre-, post- and in-order walk mean for a n-ary tree?

The tree traversal methods explained in this Wikipedia article are pre-order, post-order and in-order. Are these methods limited to binary trees? The algorithm seems to be defined in terms of left and ...
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31 views

Sort graph nodes by density [on hold]

Cyclic connected undirected graph. Every node in a graph has T value initially zero. Suppose there is a traverse via shortest path between every two nodes which increases every node's T value it ...
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36 views

Subgraph isomorphism by Ullman

I was trying to understand the subgraph isomorphism problem and I came across a slide http://oldwww.prip.tuwien.ac.at/teaching/ss/strupr/vogl.pdf In the 11'th page, the step by step description of M ...
2
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0answers
20 views

Complexity of cubic graph decomposition

I am aware that deciding the existence of decomposition of a cubic graph into edge disjoint claws is polynomial time solvable. What is the complexity of deciding the existence of decomposition of ...
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0answers
23 views

Algorithm for subgraph isomorphism [on hold]

Now I was reading up the paper of subgraph isomorphism by JR Ullman to understand the algorithm. Now I got the PDF and the algorithm looks like this: Now in step 2, what is after " if d=1 then" ? ...
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1answer
42 views

Finding common edges of two graphs

Is there any algorithm that finds the common edges and vertices between two graphs? Its not a common subgraph problem though, the edges which are common between the two graphs may not be connected to ...
2
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1answer
36 views

Proving algorithm for removing nodes from a complete graph with two kinds of edges

Lets say $G$ is complete undirected graph with a set of edges coloured either black or red. The problem is to find an algorithm answering if it is possible to remove a subset of nodes from $G$ in a ...
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2answers
43 views

Existence of shortest path in a graph with no negative cycles?

Suppose that the input graph $G$ does not have any negative cycles but however it is permitted to contain edges having negative weight. Let $s$ be the source vertex. How do I prove that for every ...
4
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1answer
99 views

Unique path sums in a DAG using vertex instrumentation

I stumbled across this paper from Ball et al. In their paper they assign specific values to the edges of a graph. When the graph is traversed, or lets call it executed (since they talk about control ...
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2answers
32 views

Finding a minimum set of dependencies in a cycle-filled dependency graph

I have a graph of a large number of targets. Each target depends on a list of other targets. The graph is very large and filled with cycles of dependencies. My goal is to find the smallest subset of ...
2
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2answers
67 views

Determine if items can be ordered grouping two simultaneous criteria

For a set of items with two properties, how can it be detemined if they can be ordered in a way so that for every value of either property all items of that value are grouped together. Obviously ...
2
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2answers
19 views

What are transitive successor and transitive predecessor in the graph?

I'm reading a book on compilers, Engineering a Compiler, 2nd ed. by Keith D. Cooper & Linda Torczon and I came a across two new terms that I can't understand, they are: transitive successor ...
2
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1answer
45 views

What is the graph with $8$ vertices and $12$ edges that has the most spanning trees? [closed]

I'm not sure if this is an open question, but what is the graph with $8$ vertices and $12$ edges that has the most spanning trees?
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0answers
21 views

Number of vertices that belong to all the maximum matchings of a graph.

The given graph is connected but not necessarily bipartite. Please describe the complete approach with useful links , I read stuff related to augmenting paths but could not comprehend well. An O(VE) ...
3
votes
1answer
50 views

How to Prove NP-Completeness of Minimum Crossing Problem?

In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. (from wikipedia) I know that the problem of counting the ...
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0answers
31 views

All paths of length n from a single graph vertex in a directed cyclic graph [duplicate]

Thanks in advance...looking for recommendations on an algorithm to find all paths of length n starting from a single node in a directed, cyclic graph. I am not concerned with at which node the path ...
2
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1answer
26 views

Intuitive idea/proof behind Kirchhoff's Matrix Tree Theorem using as little matrices/linear algebra as possible?

could someone provide me/refer me to a intuitive idea/proof behind Kirchhoff's Matrix Tree Theorem that uses as little technical details involving matrices/linear algebra as possible? I'm trying to ...
2
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1answer
50 views

Tolerated use of the term topology

In the field of data structures (and maybe in graph theory), can we use the term topology to speak about the shape of a tree? For instance, consider the two following trees : 1) The first one: Node ...
0
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1answer
41 views

Graph theory, $n$ people sitting around table [closed]

$n$ people want to have dinner together around a table for $k$ nights so that no person has the same neighbor twice. How big can $k$ be in terms of $n$? Does everybody get to sit next to everybody ...
1
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1answer
12 views

How to prove that the probability that a random graph has a stable set of size $2\lceil \log n\rceil$ is sub-constant?

Given a random graph on $n$ vertices where each edge is included with probability $1/2$. Lets call it $G=(n,1/2)$. How can we show that the probability that this graph has a stable set of size at ...
4
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0answers
71 views

Computing the “at least k friends in common” graph

Suppose we have the graph of a social network with symmetric connections (e.g. Facebook or LinkedIn). Suppose we would like to find all pairs of people who have at least k friends in common, in order ...
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1answer
40 views

SimRank on a weighted directed graph (how to calculate node similarity)

I have a weighted directed graph (it's sparse, 35,000 nodes and 19 million edges) and would like to calculate similarity scores for pairs of nodes. SimRank would be ideal for this purpose, except that ...
2
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1answer
102 views

Graph Families that are easy to color

What are the non-trivial graph families that have a known chromatic number, or an easy way (polynomial-time algorithm) to compute the latter. Examples would be: Kneser graphs Chordal graphs Do ...
0
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2answers
123 views

scale-free networks and adjacency matrix

Given a distribution over graphs with $n$ nodes having the "scale-free" property, I would like to compute for a pair of vertices $(a,b)$ the probability that they are connected (or more precisely the ...
3
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1answer
53 views

Travelling Salesman which can repeat cities

In the TSP problem, we usually assume a complete graph. If we can only visit each city once, we need a complete graph to ensure that there will be a path from every city to every other city. This is ...
0
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1answer
15 views

Necessities for two undirected graphs being isomorphic

As far as I know, for two undirected graphs $G = (V, E) $ and $H = (V', E')$, the following criteria is necessary for them to be isomorphic: $|V| = |V'|$ $|E| = |E'|$ $G$ has $j$ nodes of degree $k$ ...
3
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1answer
81 views

Locally finite graph without an optimal path

If I have a locally finite graph (every node has finite number of neighbors) with positive edge weights, is it possible for there to be a path between some start node and goal node but no shortest ...
3
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0answers
36 views

Finding partial traveling salesman path of specified length

For a given set of nodes, I can find optimal paths that visit all nodes using various traveling salesman algorithms. As a subset of this problem, I would like to be able to find shortest partial ...
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2answers
73 views

Undirected graph G that has 12 vertices, 66 edges and 3 connected components?

Why would it be impossible to draw an undirected graph G that has 12 vertices, with 3 connected components if G had 66 edges?
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2answers
61 views

Why is T not a minimum spanning tree of G?

The Problem: Let T be a tree constructed by Dijkstra's algorithm in the process of solving the single source shortest-paths problem for a weighted connected graph G.    a. True of ...
1
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1answer
57 views

How to draw a graph to disprove this statement?

The Problem: Indicate whether the following statements are true or false: a. If e is a minimum-weight edge in a connected weighted graph, it must be among edges of at least one minimum ...
0
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1answer
28 views

Algorithm deciding if a radius of a graph is two

Given an unweighted, undirected graph, what is the time complexity to decide if its radius is at most 2? Are there any faster algorithms than doing BFS on each node?
2
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1answer
83 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 $k$ ...
3
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2answers
50 views

What is the appropriate algorithm for bipartite matching with constraints?

I have a problem that is a bit complex, and I don't know what method/model I should use to express it (much less solve it). Let's say we have a lot of employees and a few jobs to be done. Each ...
2
votes
1answer
45 views

Strongly connected components on a DAG

What is supposed to be the right result of an SCC algorithm running on a DAG. should it return "no components" or "there are V components of size 1"? I suspect it will return the latter (since it ...
2
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1answer
32 views

Dominator Tree for DAG

Is there a fast algorithm to compute dominator tree for acyclic graphs? The Lengauer-Tarjan Algorithm is a fast algorithm for general flowgraphs. But if a graph is acyclic, do we have a faster ...
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1answer
18 views

Checking a property of all of the cycles in a graph

Suppose $G= (V,E)$ is a directed graph with weights on the edges. I would like to check if $G$ has the following property: if $C \subset E$ is the set of edges in a cycle of length at least $3$, then ...
3
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1answer
180 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 ...
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1answer
52 views

How can an maximum flow algorithm for directed graphs, i.e. Edmond-Karp, be adapted to compute a minimum $s$-$t$ cut in a undirected graph?

How can an maximum flow algorithm for directed graphs, i.e. Edmond-Karp, be adapted to compute a minimum $s$-$t$ cut in an undirected graph ? I've seen it stated that one can apply a maximum flow ...
2
votes
1answer
27 views

Ford-Fulkerson Running Time

This question might be really basic but every source seems to skip over a couple of steps neither of which seem trivial to me. It would be great if someone could explain them! In the analysis of ...
1
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1answer
80 views

How important is it to find a deterministic polynomial time algorithm to construct Ramanujan graphs? [closed]

As in I don't know what is the difference between say the conferences SODA, STOC or FOCS. Measured in terms of such conferences, where would such a result be publishable? This is not a "technical" ...
3
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1answer
57 views

Is the minimal number of colors needed to color a graph some fixed number?

Consider to following decision problem: Input: Undirected graph $G=(V,E)$ Question: Is the minimum numbers of colors needed to color the vertices (such that every two adjacent vertices ...
1
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1answer
32 views

Decreasing a digraph's edge-weights while keeping net weights of edges at each vertex constant

Given a directed weighted graph, is there an algorithm that does the following: Removes as many edges possible. Reduces as many weights as possible. Given the constraint that the net weight of all ...
1
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0answers
32 views

TSP heuristics for limited distance information [closed]

this is my first question on ComputerScience beta. :) I've posted a similiar question on Mathoverflow and a friendly user advised me to post my question on this site. Problem: I'm looking for ...
3
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0answers
21 views

Efficient algorithms for mutual, inverse, or round-trip Personalized PageRank

I'd like to implement a similarity between two nodes (X and Y) of a graph based on a simple extension of the Personalized PageRank algorithm, either: (Mutual PageRank): the product of the PPR of Y ...
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0answers
18 views

A way to order a shortest path tree

Given the shortest path tree of a directed graph G=(V,E) and w: E-> R , source vertex s and an assumption that there are no negative cycles in the graph. In the homework assignment we need to find ...
3
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1answer
38 views

Minimum Length Hamiltonian Path Pair in O(n^2) or better

A friend and I have been discussing turning a $O(n^2)$ graph problem's algorithm into $O(n\log n)$, or at least less than $O(n^2)$. And no - this is not a homework question. We've narrowed it down to ...
6
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2answers
115 views

Real world applications for Steiner Tree Problem?

Are there real-world applications of the Steiner Tree Problem (STP)? I understand that VSLI chip design is a good application of the STP. Are there any other examples of real world problems that ...
0
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0answers
25 views

Finding the second lightest path in a graph

Assume I have a weighted, directional graph with no cycles. What is the most efficient way to find the second lightest path from the source vertex to a given vertex?