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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6
votes
0answers
32 views

Determining the minimum number of edges to add in order to be 3-connected

A graph $G$ is said to be $3$-connected if it has no $2$-vertex cutsets (i.e., at least three vertices must be deleted to disconnect the graph). As far as I know, it is possible to determine if a ...
1
vote
2answers
35 views

Can't see how this can be true: any connected undirected graph $G$ contains vertex v such that removing v results in another connected Graph $G'$ [on hold]

I am attempting to prove this result but I found a case where I can easily disprove this statement. Suppose $G$ is a graph with two nodes u, v and an edge (u,v) and respective self edges, then ...
4
votes
1answer
49 views

Distribution of cycles length in a graph

Given a random directed Graph G: $$ G=(V,E) \\ \lvert V \rvert = n , \lvert E \rvert = k $$ where for each vertex, either: $$ d_{incoming}(v) = 1 , d_{outgoing}(v) = 1 $$ meaning - for each ...
-1
votes
0answers
16 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
0answers
18 views

Necessary and sufficient condition for unique minimum spanning tree

This is an exercise problem (Ex.3) from the excellent lecture note by Jeff Erickson Lecture 20: Minimum Spanning Trees [Fa’13] . Prove that an edge-weighted graph $G$ has a unique minimum ...
2
votes
2answers
39 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 ...
-1
votes
1answer
25 views

Vertex degree in De Bruijn graphs

How can I find the degree of a vertex in a De Bruijn graph? Also, given a vertex, is there an easy way to find which other vertices it is connected to?
2
votes
1answer
36 views

Assign undirected edges in a mixed graph to make graph cyclic/acyclic

What is the complexity of the following problem? Given a mixed (some edges directed, some undirected) graph, assign a direction to all the undirected edges to make the graph contain a cycle. It ...
4
votes
2answers
209 views

Constructing a random Hamiltonian Cycle (Secret Santa)

I was programming a little Secret Santa tool for my extended family's gift exchange. We had a few constraints: No recipients within the immediate family Nobody should get who they got last year The ...
0
votes
0answers
13 views

Prove a characterisation of the minimum 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 ...
3
votes
1answer
78 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 ...
3
votes
0answers
45 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 ...
7
votes
2answers
274 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
55 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
1answer
35 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
63 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 ...
5
votes
0answers
79 views

Minimal polynomial reduction of dominating set to max clique [migrated]

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
20 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
120 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
votes
0answers
26 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 ...
0
votes
1answer
46 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
vote
0answers
24 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
51 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
54 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
votes
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
16 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
26 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
46 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
votes
0answers
30 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
votes
2answers
72 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
votes
0answers
27 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
74 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
26 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
101 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
votes
1answer
51 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
422 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
32 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
63 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
120 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
42 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
76 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
38 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
48 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
44 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
votes
0answers
222 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
59 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
votes
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 ...