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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2
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1answer
24 views

TSP polynomial Time

How can it be proved that TSP cannot be solved in polynomial time ( Please bear that I don't have a hardcore computer science background).
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0answers
12 views

What are the known NP-hardness or optimization results about spectrum of matrices? [on hold]

Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are some known optimization results about their possible spectral radius or spectral gap? [..I am calling a symmetric matrix to be ...
1
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1answer
14 views

Solving cycle in undirected graph in log space?

Setting Let: $$UCYLE = \mathcal \{ <G> ~:~ G \text{ is an undirected graph that contains a simple cycle}\}.$$ My Solution we show $UCYLE \in L$ by constructing $\mathcal M$ that decides ...
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0answers
25 views

$G$ has an Euler tour iff in-degree($v$)=out-degree($v$)

A simple cycle is a path in a graph that starts and ends at the same vertex without passing through the same vertex more than once. A complex cycle, is a cycle that passes through the same vertex ...
2
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1answer
22 views

Partitioning planar graph cycles based on chords

Given a cycle of length > 3 in a planar graph, I'm looking to partition it into 4 sublists of length 2 or more such that: No sublist contains two vertices with a chord between them The last element ...
1
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0answers
28 views

Terminology for a graph with ports on its nodes

A Graph is a well-defined concept in mathematics, computer science and engineering disciplines that depend on them. However, oftentimes a practical implementation of a (directed) graph in a certain ...
4
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1answer
102 views

Gray Code Generation

I am trying to generate a $n$-bit gray code where I can specify two strings $s$ and $t$ that must be consecutive in that gray code. I know that there are ways to generate specific codes (such has the ...
4
votes
1answer
79 views

Amplifying the correctness of $\mathsf{RP}$ algorithms using expander graphs

A graph $G = (V, E)$ is called an $(n, d, \varepsilon)$-expander if the graph has $n$ vertices, maximum degree $d$, and satisfies the following expansion property: for every subset $W\subset V$ such ...
2
votes
1answer
54 views

Shortest paths in weighted graphs, and minimum spanning trees

I stuck in one challenging question, I read on my notes. An undirected, weighted, connected graph $G$, (with no negative weights and with all weights distinct) is given. We know that, in this ...
2
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1answer
29 views

Route planning for a car driver picking up people using public transport

We just had an interesting though for a routing algorithm for people carpooling. Imagine the following situation: Person 1 is driving with his car from the south of city A to city B far in the north. ...
2
votes
1answer
71 views

Lowest Common Ancestor from children up?

I've seen algorithms for finding the lowest common ancestor from the root of a tree. However, I'm interested in finding the LCA of two distinct Nodes in a (not necessarily binary) tree from the bottom ...
0
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1answer
55 views

Minimum size of largest clique in graph

I'm having trouble with a problem from HackerRank, and I'm hoping someone here can enlighten me. The problem is stated like this: What is the minimum size of the largest clique in any graph with N ...
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0answers
20 views

Graph Centrality: spectral techniques

What is the difference between: normalizing the row of an adjacency matrix and taking the right eigenvector normalizing the row of an adjacency matrix and taking the left eigenvector normalizing the ...
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0answers
21 views

Betweenness centrality and least average shortest path

TL;DR: How do I justify mathematically that vertices with the highest betweenness centrality do not necessarily have the smallest mean shortest path? I am currently studying the London Underground as ...
0
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1answer
44 views

Matching a set of paths to an incrementally generated graph

I am working on an approximate matching problem, where I have a set of paths in an unknown graph (A) and a partial graph (B), where B is generated incrementally during the matching process (and can be ...
2
votes
1answer
61 views

Relationship between Independent Set and Vertex Cover

Directly from Wikipedia, a set of vertices $X \subseteq V(G)$ of a graph $G$ is independent if and only if its complement $V(G) \setminus X$ is a vertex cover. Does this imply that the complement of ...
-1
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1answer
47 views

Comparing two graphs, finding vertices that changed their positions

I have a task of comparing two organisation charts. These chart objects are described as a set of nodes (people) where each has a unique ID field and a parent ID field (pointing to another node's ...
5
votes
1answer
38 views

Heaviest planar subgraph

Consider the following problem. Given: A complete graph with real non-negative weights on the edges. Task: Find a planar subgraph of maximum weight. ("Maximum" among all possible planar subgraphs.) ...
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0answers
37 views

Diameter and some Formula on Graph Theory

in an undirected graph G, we define: Diameter is maximum of minimum paths between two vertex of a graph. L(s) is maximum length of minimum paths from s to ...
2
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0answers
43 views

Pagerank is equivalent to degree centrality

Can someone explain why pagerank defined for undirected graph with no damping factor is equivalent to the degree? $\sum_{j\in N(i)}{\frac{p(j)}{d(j)}} = d(i)$ I looked up every book I could, but ...
0
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1answer
43 views

Christofides algorithm: why must an MST have even number of odd-degree vertices?

This question is not necessarily related to Christofides algorithm per se, I just ran into it when reading about it. I assume that a minimum spanning tree must have an even number of odd-degree ...
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0answers
32 views

Lower bounding the minimum equivalent graph

The transitive reduction $G^t = (V,E^t)$ of a graph $G=(V,E)$ is the smallest graph with the same reachability as $G$ with the property $E^t \subseteq V \times V$. The minimum equivalent graph $G' = ...
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0answers
27 views

Is the weighted transitive reduction problem NP-hard?

The transitive reduction problem is to find the graph with the smallest number of edges such that $G^t = (V,E^t)$ has the same reachability as $G=(V,E)$. When $E^t \subseteq E$ it is NP-complete. ...
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0answers
64 views

Totally unimodular <=> polynomial time?

Crossposting due to recommendation. I formulated a MIP problem which I didn't expect to be unimodular. The problem is to find a minimum complete sequence in a strongly connected digraph. That is, ...
3
votes
1answer
70 views

Is a “tree” with $0$ vertices, $0$ edges or $1$ vertex, $0$ edges considered a valid tree?

For the following $2$ cases: (1) $V = \emptyset, E = \emptyset $ (i.e. nothing at all) (2) $V = \{v_0\}, E = \emptyset $ (i.e. only 1 root node $v_0$) Are they considered a valid tree? It seems ...
4
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1answer
111 views

Finding all circuits that contain a given edge

Given a directed graph $G = (V, E)$ and an edge $e \in E$, I'm trying to come up with an algorithm to construct the minimum induced subgraph $H$ of $G$ with the property that every circuit in $G$ that ...
3
votes
1answer
46 views

Sufficient condition for simple graph isomorphism?

Say I have two simple graphs, $A$ and $B$ In $A$, I know: one node has 3 nodes at distance of 1, 4 nodes at distance 2, etc. one node has 4 nodes at distance of 1, 1 nodes at distance 2, etc. etc. ...
0
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1answer
61 views

Is it possible to convert a graph with one negative capacity to a graph with only positive capacities?

I am interested in whether a graph (say, a complete graph) with one capacity negative (or many, but one should suffice) can be reconstructed as a graph with all non-negative capacities where the max ...
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0answers
19 views

Understanding the correctness of the Euler Tour Technique

I can't prove the correctness of the following algorithm by R. E. Tarjan amd U. Vishkin, as described on wikipedia: Given an undirected tree presented as a set of edges, the Euler tour ...
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0answers
18 views

Stable matching of producers, consumers and objects

Has the following version of the stable matching problem been studied? There are $k$ types of objects. There are $n$ producers, each of whom can produce a single object of any type, and has a ...
0
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1answer
39 views

undirected graph without weights and DFS [closed]

following question on undirected graph without weights can be solved by using DFS and in O(|V|+|E|) times. check that G is ...
0
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2answers
93 views

Finding a Hamiltonian Path through the complete graph on 37 vertices: $K_{37}$ [closed]

I'm planning on making a fiber art $K_{37}$ (like the one I laser etched with help: K37: The complete graph on 37 nodes, svg). To accomplish this, the plan is to construct 37 pegs equally spaced in a ...
5
votes
1answer
139 views

Generate scale-free networks with power-law degree distributions using Barabasi-Albert

I'm trying to reproduce the synthetic networks (graphs) described in some papers. It is stated that the Barabasi-Albert model was used to create "scale-free networks with power-law degree ...
2
votes
1answer
31 views

Examples for directed graphs with super polynomial cover time

The cover time of a graph is the expected number of steps in a random walk on the graph until we visit all the nodes. For undirected graphs the cover time is upperbounded by $O(n^3)$. What about ...
3
votes
1answer
122 views

Equivalent Straight Line Embedding of a Planar Graph Drawing on a Grid

An embedding of a graph G on a surface Σ is a representation of G on Σ in which points of Σ are associated to vertices and simple arcs are associated to edges in such a way that: the endpoints of ...
0
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1answer
36 views

Not Hamiltonian is in NP Class? [duplicate]

I ask a question before, Questions on Graph and Hamiltonian, but i ask it here with different challenging contest. From this book and other study in complexity theory, I have seen the following ...
4
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0answers
35 views

Parallel algorithm to find if a set of nodes is on an elememtry cycle in a directed/undirected graph

I'm looking to find / develop a simple parallel algorithm that does this: Input: vs: list of root vertices max_length: max cycle length max_dist: max distance to root Variants one variant of ...
2
votes
1answer
79 views

Questions on Graph and Hamiltonian [closed]

From this book and other study in complexity theory, I have seen the following statement: The definition of NP is not symmetric with respect to yes-instances and no-instances. For example, it is ...
2
votes
1answer
34 views

What is an edge hop?

I've tried googling it, but found nothing. Here is the context it's in: From Bayesian Reasoning and Machine Learning: Adjacency matrices may seem wasteful since many of the entries are zero. ...
2
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0answers
23 views

Heuristic for weighted maximum independent set in graph with ~$2 \times 10^5$ nodes and $|E| \propto |V|$

I want to find a near-optimal solution for a maximum weight independent set. i.e given a graph $G = (V,E)$ I want to find a set $S = \{v_1,v_2,\dots,v_n\}$ of nodes in $V$ such that the sum of their ...
4
votes
1answer
81 views

Term for a matching which is perfect on one side only

What is a standard term for a matching in a bipartite graph, in which one part has less vertices than the other part, and the part with less vertices is fully matched (but the other part is, ...
0
votes
0answers
12 views

Solving $Isomorphism$ using $AUTOM$ in polynomial time

Let $Iso$ be the language of all $<G,H>$ such that $G$ and $H$ are isomorphic, and $AUTOM$ be the language of all $G$'s such that $G$ has a non-trivial automorphism. I'd like to show that, ...
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votes
2answers
34 views

A* graph search heuristicfor pathfinding

A* needs a consistent heuristic to work on a graph. So I'm not sure if the heuristic of a straight line (bird flight) can be used. For example: the costs to travel to a neighbors node is always ...
0
votes
1answer
33 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 ...
2
votes
1answer
56 views

Reduction from a further constrained problem

If I find an NP Hard problem that is equivalent to my problem with an additional constraint or bound, can I still prove that my problem is NP Hard? Generally, this is probably not the case. For ...
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3answers
98 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, ...
2
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0answers
61 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 ...
3
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1answer
100 views

Linear programming formulation of cheapest k-edge path between two nodes

Given a directed graph $G = (V,E)$ with positive edge weights, find the minimum cost path between $s$ and $t$ that traverses exactly $k$ edges. Here is my attempt using a flow network: \begin{align} ...
2
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0answers
66 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 ...
4
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0answers
61 views

Recognize if graph has Hamiltonian cycle from subgraphs

There is a graph G, which is not known to me. Instead I am given the multiset of all graphs that are obtained by deleting a single vertex from G. My task is to figure out, from all of these subgraphs, ...