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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0
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1answer
20 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
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1answer
27 views

Single Source Shortest Path: What does the weights on the vertex and edges tell you?

In MIT's open courseware (http://courses.csail.mit.edu/6.006/spring11/lectures/lec15.pdf), I do not see how computing a set of numbers on the edge and the vertex will produce the shortest path. ...
2
votes
1answer
51 views

Need a hint! Karger's algorithm versus Kruskal, spanning tree distribution

Let G = (V,E) be a unit-capacity graph with n vertices and m edges. Let T denote all the spanning trees in G. If we run Karger's algorithm, we will get a random spanning tree in T formed by the ...
-3
votes
0answers
26 views

how to reduce 3-colorable graph to this? [on hold]

suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best to reduce to ...
4
votes
0answers
61 views
+100

Minimal polynomial reduction of dominating set to max clique

Let $G$ be a simple undirected graph. Recall that $S \subseteq V(G)$ is a dominating set of $G$ if every vertex of $v \in V(G) \setminus S$ has a neighbour in $S.$ It is well known that it is NP ...
1
vote
1answer
17 views

Construction of graph with given Wiener Index

Given the sum of weights of shortest paths between all vertices in a graph, how can I construct a connected graph that satisfies the given sum? That is, how can a graph with a given wiener index be ...
0
votes
2answers
117 views

Show that the tree resulting from BFS is a spanning tree?

Given that $G$ is some connected and undirected graph, and I want to run BFS on it from some starting vertex. How can I show that $T = \{ \{\text{predecessor}[u], u\} \mid u \text{ is a vertex}\}$ is ...
2
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0answers
21 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 ...
-1
votes
1answer
44 views

Calculating genus of graph

How to calculate genus of arbitrary graph? I am interested in any algorithm, even it based on full search.
1
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0answers
22 views

Efficient flood filling (seed filling)

I am referring to the algorithm that fills a white area of arbitrary shape in a binary digital image, starting from a given white pixel, using the Moore (8 neighbors) or Neumann (4 neighbors) ...
2
votes
2answers
46 views

Is single-source single-destination shortest path problem easier than its single-source all-destination counterpart?

Dijkstra's algorithm (wiki) and Bellman-Ford (wiki) algorithm are two typical algorithms for the single-source shortest path problem. Both of them compute distances for all nodes from source $s$. ...
2
votes
1answer
41 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 ...
0
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0answers
32 views

Cycles in graphs with optional edges, redux: labelled optional edges

In a previous question, I asked how much information is needed to encode the possible cycles in a directed graph with $N$ "optional" edges given only the subset of the optional edges that are present ...
0
votes
0answers
15 views

Assign colors to vertices of a 3 colorable graph [duplicate]

Black Box tell is g(v,e) is 3 colorable or not. How can we use this graph to assign the colors to a 3 colorable graph?
1
vote
0answers
15 views

Can independence numbers of box products of cycles increase after stabilizing?

Is there an evidence or a proof that the independence of strong products of graphs can increase after stabilizing? I am interested in odd cycles only. Let $C_n$ be an odd cycle and $\alpha(G)$ ...
0
votes
1answer
20 views

How do deal with the following situations using Prim's algorithm?

Consider the following Graph We want to generate the MST using Prim's algorithm. Starting from node A, suppose we pick B as our next node, we see a self-loop that has less weight than the two other ...
2
votes
1answer
42 views

Characterizing cycles in a directed graph with optional edges

Consider a directed graph in which some edges are marked as "optional". A graph with $N$ optional edges induces a family of $2^N$ graphs depending on which edges are removed. In some cases, some of ...
0
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0answers
21 views

What is an example of a minimum weight connected subset T of edges from a weighted connected graph G?

What is an example of a minimum weight connected subset T of edges from a weighted connected graph G? Can I just take two edges that are connected and have min combined weight and call that the min ...
2
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2answers
68 views

Is there a difference between perfect, full and complete tree?

Is there a difference between perfect, full and complete tree? Or are these the same words to describe the same situation?
0
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0answers
24 views

Merging two disconnected graphs

Firstly, I'd like to apologize for any misused terms or ways I could have made the description much more succinct. It's been a while since I took machine learning during my bachelor's. I have two ...
4
votes
3answers
67 views

Is the height of the tree the number of edges or number of nodes?

I'm so confused by some of the theorems online about tree heights. Does tree height mean the number of edges or nodes? if nodes, does it include the node it is counting from? Can the height of a tree ...
0
votes
1answer
23 views

2-way Graph Partitioning problem

We have a graph $G=(V,E)$ and we need to divide this graph into two clusters $A$ and $B$. Some pairs of vertices $u$, $v$ should not be in the same cluster, and we define an edge $(u,v) \in E$. The ...
2
votes
1answer
59 views

Acyclic Graph in NL

From the book The Nature of Computation by Moore and Mertens, exercise 8.9: Consider the problem ACYCLIC GRAPH of telling whether a directed graph is acyclic. Show that the problem is in NL, and ...
-1
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1answer
50 views

Relative Importance in Graph Theory

I am working on an algorithm that ranks a set of nodes in a graph with respect to how relative this node is to other predefined nodes (I call them query nodes). The way how the algorithm works is ...
5
votes
1answer
406 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?
-1
votes
1answer
30 views

Determine whether there is a valid rounding in a table of numbers

I was told this question would be better suited here: Suppose you have a table such as: $\begin{array}{ccc} 11.998 & 9.083 & 2.919 &|& 24\\ 12.983 & 10.872 & 3.145 ...
1
vote
0answers
62 views

Check whether a directed, rooted spanning tree is actually some shortest-paths tree in $O(V + E)$ time

Given a directed graph $G = (V, E)$, with all edge weights being non-negative, someone has written a program that he/she claims implements Dijkstra's algorithm. For a fixed starting vertex $s$, the ...
2
votes
2answers
117 views

Lovasz theta of even cycle

How does one show Lovasz theta of even $n$-cycle ($n$ is even) is of form $\frac{n}{2}$? Why is the Lovasz theta of such cycles not of form $\frac{n \cos(\frac{\pi}{n})}{1+\cos(\frac{\pi}{n})}$. Could ...
0
votes
0answers
39 views

Remove atmost K subtrees

Given a tree with N vertices numbered from 1 to N. The vertex 1 is the root of the tree. Each vertex is assigned with an integer weight. A remove operation can remove sub-tree rooted at an arbitrary ...
4
votes
1answer
74 views

Maximizing sum of ranges of vertices edges

Consider an arbitrary undirected graph $G = (V,E)$. Suppose you have a collection of $|E|$ positive integers and each integer must be assigned to one edge. Let us denote the collection of integers ...
-1
votes
1answer
36 views

Understanding A* Search on Tropical Island

I am working on an online course on AI and I am now working to understand A* better. Basically, right now I am working on a problem where: we live on a tropical island and we're trying to navigate ...
0
votes
1answer
42 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
2answers
40 views

Proving that shortest path distance of adjacent nodes can't differ by more than one

Could someone explain this proof to the following question? Lemma 22.1 from intro to algorithms Let $G=(V,E)$ be a directed or undirected graph, and let $s\in V$ be any vertex. Then, for any ...
2
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0answers
219 views

Showing that the language of graphs and nodes on an odd cycle is in NL

Let L be the language containing all the pairs (G,v) where G is a directed graph and v is a vertex in G such that G contains a cycle that contains v and the number of different vertices that appear ...
1
vote
0answers
22 views

Why is it that in a butterfly network, there is a unique path from the input to the output?

Consider the a butterfly network as defined on the following OCW notes on page 208. An explantation of it can also be found on the following page. I was wondering if someone had a proof or an ...
1
vote
2answers
57 views

For a graph to be connected, you need at least n-1 edges rigorous proof

This fact seems obvious but I was unsure how to go about proving it very rigorous. Let $|V| = n$ and $|E| = m$ for some connected graph $G$. Then consider the following proposition: If a graph is ...
0
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2answers
43 views

Does 2-edge-colourability imply 2-colourability?

Why is it that if the edges of an undirected graph G can be grouped into two sets such that every vertex is incident to at most 1 edge from each set, then the graph is 2-colorable. The reason that I ...
1
vote
1answer
17 views

Why do Benes networks form bipartite graphs when you build a constraint graph for them?

I was learning about Benes networks and was wondering why they formed bipartite graphs (and thus are two colorable) when one draws a constraint graph for them. The constraint graph is based on the ...
4
votes
1answer
82 views

Decide whether there exists a walk of weight exactly k

Consider the following problem: Input: a directed graph $G = (V,E,\omega)$ where $\omega : E \longrightarrow \mathbb{Z}$, two vertices $v_1, v_2 \in V$, and a weight $k \in \mathbb{Z}$ Question: ...
1
vote
1answer
63 views

Prove any binary tree with $n$ nodes has at least $1+\log_2 n$ levels

Prove that any binary tree with $n$ nodes has at least $1+\log_2 n$ levels. I tried setting $n=8$ and plugging in $8\geq\log_2 8 = 8\geq 3$. But I'm not sure how I can prove this by induction.
1
vote
1answer
24 views

How does the Vertex Cover algorithm by Chen et al find its tuples?

I'm still fighting with the aforementioned paper "Improved upper bounds for vertex cover" by Chen, Kanj, Xia (PDF kindly provided by Yuval Filmus). My current problem is that it's specified that the ...
3
votes
1answer
43 views

Finding vertices for which there either exists a path to all other vertices or other vertices have a path to them

Or in other words, find all $v \in V$ such that there exists a path $\forall w \in V$ $v \rightarrow w$ or $w \rightarrow v$. This is for a directed acyclic graph. I need to find an $O(|E| + |V|)$ ...
1
vote
1answer
152 views

Is finding if a graph has k isolated nodes a NP-Complete problem?

I was wondering if finding if a graph has k or more isolated nodes is a NP-Complete problem. I found the following problem: Prove that the following problem is NP-Complete. Given a set of T ...
2
votes
2answers
112 views

Finding an exactly weighted st-path in a digraph

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
1
vote
1answer
41 views

Betweenness Centrality measurement in Undirected Graphs

I'm working with graphs of a very large size (> 60k vertices), and want to speed up B.C. measurements. It is defined here: http://en.wikipedia.org/wiki/Betweenness_centrality The algorithm that I am ...
6
votes
1answer
69 views

Implementing general vertex folding procedure in an undirected graph

I'm implementing the algorithm presented in "Improved Parameterized Upper Bounds for Vertex Cover" paper (PDF). I'm a bit stumped by the General-Fold procedure. ...
3
votes
1answer
68 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 ...
3
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0answers
60 views

What is this discrete/combinatorial optimization problem?

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. ...
7
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0answers
106 views

Change in the distances in a graph after removal of a node

Given an undirected unweighted graph $G=(V,E)$ and a node $s \in V$, we are looking for a vector $\operatorname{diff}[]$, such that, $$\operatorname{diff}[v] = \sum_{u \in V \setminus \{v\}}{(d^{G ...
0
votes
0answers
96 views

Dynamic distance from source in a directed graph (only incremental or only decremental)

At the beginning we have a directed unweighted graph of $n \leq 10^3$ vertices, and $m \leq 10^5$ edges, with some vertex being a source, and we perform updates and queries on it. An update is adding ...