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
26 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).
0
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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 ...
2
votes
1answer
80 views

“Minimum” maximum flow with extra capacities

Problem: Suppose there is a graph, a source and a sink. Each edge has a capacity and an extra capacity that it can hold. If sink needs a defined amount of flow F, ...
-1
votes
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 ...
1
vote
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 ...
2
votes
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
vote
0answers
29 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
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 ...
4
votes
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 ...
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 ...
3
votes
3answers
639 views

Does spanning tree make sense for DAG?

Why cannot I find any information about spanning tree for DAG ? I must be wrong somewhere.
12
votes
2answers
2k 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 ...
2
votes
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. ...
3
votes
1answer
42 views

Common subgraph isomorphism with K vertex

I'm looking for subgraph isomorphism of at least K vertex between Graph A and B. I only can come up with the dumbest algorithm, which is: Compute all combination of vertices with length K of Graph ...
2
votes
1answer
54 views

Question about spanning trees and creating them through BFS and/or DFS algorithms

The question is as follows: True or False: For every non-directed connected non-weighted graph and for every spanning tree T of the graph there exists a vertex v such that T is a DFS tree with the ...
6
votes
1answer
683 views

Bellman-Ford variation

I have a "smarter" version of Bellman-Ford here; this version is more clever about choosing the edges to relax. ...
8
votes
1answer
530 views

Does a graph always have a minimum spanning tree that is binary?

I have a graph and I need to find a minimum spanning tree to a given graph. What is to be done so that the output obtained is a binary tree?
0
votes
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 ...
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 ...
1
vote
1answer
74 views

Union grouping in bipartite graphs?

I'm trying to figure out a good (and fast) solution to the following problem: I have two roles I'm working with, let's call them players and teams having many-to-many relationship (a player can be on ...
0
votes
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 ...
6
votes
1answer
426 views

Why is the complexity of negative-cycle-cancelling $O(V²AUW)$?

We want to solve a minimal-cost-flow problem with a generic negative-cycle cancelling algorithm. That is, we start with a random valid flow, and then we do not pick any "good" negative cycles such as ...
9
votes
2answers
226 views

The equivalence relations cover problem (in graph theory)

An equivalence relation on a finite vertex set can be represented by an undirected graph that is a disjoint union of cliques. The vertex set represents the elements and an edge represents that two ...
-1
votes
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 ...
1
vote
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
votes
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 ...
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.) ...
-3
votes
0answers
38 views

Diameter and some Formula on Graph Theory [on hold]

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
votes
0answers
123 views

Travelling salesman problem with detours

I am interested if there exists a following version of the travelling salesman problem: INSTANCE: A finite set $C = \{1,2,\dots,k\}$ of cities, a positive integer distance $\delta(i,j)$ for each ...
5
votes
2answers
272 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 ...
2
votes
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
votes
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 ...
0
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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. ...
1
vote
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' = ...
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 ...
1
vote
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
92 views

Disconnecting a complete graph by removing edges randomly

Given a complete graph with $n$ nodes, I remove edges randomly with probability $p$ such that I want to disconnect the graph. I want to find out the minimum number of edges that I must remove ...
3
votes
1answer
85 views

How do we know to what community a vertex belongs to in the Girvan-Newman algorithm?

So I've been doing some reading on community detection in graphs as I'm planning on working on my thesis for it. I've been reviewing papers regarding the same and came across the Girvan-Newman ...
4
votes
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 ...
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 ...
0
votes
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 ...
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. ...
2
votes
2answers
78 views

Embedding a general planar graph into a grid

I have here a little problem with my homework, and would appreciate some direction. I am attempting for some time now to show that every planar graph is embeddable into a grid (As large as needs be). ...
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 ...
1
vote
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 ...
1
vote
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
votes
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 ...
0
votes
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 ...
4
votes
0answers
80 views

What is this prize-collecting optimization problem with travel times?

There exist very rich literature on discrete optimization problems such as variants of knapsack problem, traveling salesman problem, orienteering problem, tourist trip design problem and etc. ...