Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,490
questions
0
votes
0
answers
36
views
Are there good analogues to Sparsest Cut/Balanced cut for vertex separators instead of edges cuts?
Most problems about cutting graphs into roughly equal parts such as Sparsest cut, Graph partition, Balanced Cut, etc are based on minimizing the size of an edge cut. Even if all of those problems are ...
-2
votes
0
answers
27
views
Comparing objects [closed]
Say for example you have 100 list objects, each list containing a mixture of strings and integers, contained within a dictionary like so
{a : [100, "cat", 50, "mouse"], b : [100,&...
- 1
0
votes
0
answers
23
views
Comparing communication network graphs [closed]
I started out with a grid graph,
performed some operations on it, and ended up with a set of networks; for example,
,
,
,
I need to compare these graphs.
A thought that I had was to compare them with ...
- 101
2
votes
1
answer
69
views
Regularity Lemma for Multi-Relational Graphs?
Is there an analogous to Szemerédi regularity lemma in the setting, where I have multi relational graph i.e. I have $n$ nodes, but instead of having edges to be in $\{0,1\}$ i.e. there is an edge or ...
- 704
0
votes
0
answers
37
views
Comparing networks using graph theory [closed]
I'm new to graph theory so forgive if I use unconventional terminology. Please ask if there's any confusion regarding the statements I make.
I have a bunch of undirected, unweighted, simple graphs ...
- 101
2
votes
0
answers
69
views
Finding Hamilton cycles in random graphs
For a random graph $G$ of minimum degree 3, can we find a Hamilton cycle in linear time (with high probability for every edge density)?
If this is an open problem, I will also accept an empirically ...
- 2,362
1
vote
0
answers
49
views
Find the minimum cost spider joining a root to some leaves
A spider is a tree with at most one vertex of degree greater than 2. This vertex is called the head of the spider.
I am interested in the following problem: We are given an undirected graph $G = (V,E)$...
- 567
0
votes
0
answers
25
views
How to maximize flow through a graph based on edge orientation (in 3D Cartesian Coordinate Space)?
Problem Stmt:
Suppose you have a graph $G$ with edges $E$ and nodes $V$. The nodes have ${x,y,z}$ coordinates in 3D Cartesian space. Assuming each node contains an $x$ amount of material, the idea is ...
- 1
2
votes
0
answers
64
views
Origin of Berge's (Weak) Perfect Graph Conjecture
In an account of his thought process (refer p. 3) leading up to the perfect graph conjecture (which I'm preparing a seminar talk on), C. Berge states what seems to be a crucial step:
(1) a graph $G$ ...
- 21
0
votes
0
answers
68
views
What's the difference between 'theoretical' and 'applied' runtime complexity?
I recently followed a course on complexity theory and according to the professor I got one particular question wrong. The question involved the runtime complexity of checking the correctness of a ...
- 1
1
vote
0
answers
48
views
The "branch-depth" parameter and its use in FPT algorithms
Let $P=(v_1,\dots,v_q)$ be an induced path in the undirected graph $G(V,E)$. In [1], the authors define the branch depth of $P$ to be $b(P)=|N_G[\{v_1,\dots,v_{q-1}\}]|-1$. Further, in [1] it is shown ...
- 75
1
vote
0
answers
52
views
The tree augmentation problem, but with hyperlinks
In the (Weighted) Tree Augmentation Problem, we are given a tree $T = (V,E)$ and a set of additional edges $L$ called links with non-negative costs. Each link $\ell = (u,v)$ covers the tree edges ...
- 567
0
votes
1
answer
78
views
Is there FPT or XP algorithms known for Shortest Steiner cycle and $(a,b)$-Steiner path problem
Shortest Steiner cycle and $(a,b)$-Steiner path problem are generalizations of optimization versions of Hamiltonian cycle and Hamiltonian path problems.
The Shortest Steiner cycle problem is defined ...
- 141
1
vote
0
answers
50
views
Is there an FPT or XP algorithm known for this version of $k$-edge disjoint paths problem?
The shortest $k$-edge disjoint paths problem is defined as follows:
Input: An undirected graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$.
Question: Find (if exist) $k$-pairwise ...
- 141
1
vote
0
answers
76
views
Cheapest Insertion is $2$-approximation for TSP
Consider the Cheapest Insertion Algorithm on a complete graph with $n$ vertices, where each edge $uv$ has a weight $w(uv)$, and the weights satisfy the triangle inequality $w(xz)\leq w(xy)+w(yz)$ for ...
- 11
0
votes
0
answers
91
views
Separation oracle for breaking cycles in directed graph
I am working on a directed graph problem and am collaterally interested to know whether there is a separation oracle for the following set of linear constraints.
We are given a directed graph $G$ ...
- 89
0
votes
0
answers
35
views
Is there an MMSNP formula for 3-colouring?
By MMSNP, I mean Monotone Monadic SNP without inequality. For $k\in\mathbb{N}$, the problem $k$-COLOURABILITY takes a graph $G$ as input and asks whether $G$ is $k$-colourable.
It is well-known that ...
- 1,623
1
vote
0
answers
15
views
Locally bijective homomorphism between locally-H graphs
Graphs in this question are finite, simple and undirected.
For a fixed graph $H$, a graph $G$ is said to be locally-$H$ if for every vertex $v$ of $G$, the neighbourhood of $v$ in $G$ induces $H$ (i.e....
- 1,623
2
votes
0
answers
62
views
Is k-ACYCLIC COLOURABLITY in CSP?
All graphs in this question are finite, simple and undirected.
Let $k$ be a fixed positive integer.
A $k$-colouring of a graph $G$ is a function $f\colon V(G)\to\{1,2,\dots,k\}$ such that $f(u)\neq f(...
- 1,623
4
votes
1
answer
84
views
Tensor network contraction "bubbling": why are some approaches more computationally efficient than others (question from a beginner)
I am learning the very basics of tensor network theory and I am trying to understand why some ways to contract tensors are better than others in terms of computational complexity. Knowing in which ...
0
votes
0
answers
49
views
Locally bijective homomorphism between line graphs
Graphs in this question are finite, simple and undirected.
A function $\psi\colon V(G)\to V(H)$ is a locally bijective homomorphism from $G$ to $H$ if for every vertex $v$ of $G$, the restiction of $\...
- 1,623
0
votes
0
answers
66
views
Does the Christofides algorithm ensure this inequality?
Let $(X,d)$ be a finite metric space. Let $C$ be a Hamiltonian cycle (over $X$) outputted by Christofide's algorithm. Also, let $K$ be a minimum spanning tree. I am aware that Christofide's algorithm ...
- 141
0
votes
0
answers
34
views
On infinite bipartite graphs with no infinite rays
I'm working on a problem in logic and it reduces to proving that a certain infinite graph is connected. The graph has the following properties:
It is bipartite
It is not necessarily finite (or ...
- 193
0
votes
0
answers
19
views
2-connectivity of dual of a minimal cut in a bounded genus graph
Let $G$ be a graph of genus $g$ embedded on a surface of genus $g$. Let $s,t \in V(G)$. Consider a minimal $s,t$-cut $C$ in $G$. Let $H$ consist of the union of faces adjacent to $E(C)$. Notice that $...
- 2,267
2
votes
1
answer
116
views
Is the center of a BFS tree a good approximation of the graphs center?
Given a graph $G=(V,E)$, a center is a vertex $v\in V$ with minimal eccentricity (i.e., $v\in\text{argmin}_v\max_u d(u,v)$).
Finding the center of the graph can easily be done using all-pairs-shortest-...
- 21
2
votes
1
answer
107
views
Proving a property of minimal st-separators that are not minimum st-separators
Let $G$ be an undirected, connected graph, and $s,t$ non-adjacent vertices in $G$.
Denote by $k_{st}(G)$ the $st$-connectivity of $G$. That is, $k_{st}(G)$ is the size of any minimum $st$-separator of ...
- 75
3
votes
3
answers
475
views
Evaluating asymptotic probabilities of First Order Logic Formulas?
0-1 Laws in first order logic state that the probability of a FOL sentence $\Phi$, defined as follows:
$$P(\Phi) = \frac{|\{\omega \in \Omega^{n}:\omega \models \Phi\}|}{|\Omega^{n}|} $$
where $\Omega^...
- 704
1
vote
0
answers
24
views
Upper Bound for distance-two chromatic number in terms of maximum degree
Let us consider simple,finite, undirected graphs. A distance-two colouring of a graph $G$ is a fuction $f:V(G)\to\{1,2,\dots\}$ such that $f(u)\neq f(v)$ whenever $dist_G(u,v)\leq 2$. A distance-two ...
- 1,623
0
votes
0
answers
38
views
Reducing computing the partition function to computing the number of min-cardinality (s, t) cut
Consider a partition function for a graph as follows:
\begin{equation}
\mathrm{Z}_\mathrm{G}(\beta) = \sum_{z \in \{-1, 1\}^{n}} \beta^{\underset{(i, j) \in E, i < j}{\sum} w_{i,j} ~z_i z_j},
\end{...
- 153
7
votes
2
answers
196
views
Directed acyclic graphs with logarithmic diameter
Fix an ordering $v_1,\ldots, v_n$ of the vertices $V$ of a directed acyclic graph (DAG), so if there is a directed edge from $v_i$ to $v_j$ then $i < j$. Define the diameter of the graph to be the ...
- 201
1
vote
0
answers
64
views
On Negami's planar cover cojecture
For this question, let us consider only simple, finite, undirected graphs.
A homomorphism $\psi$ from a graph a $G$ to a graph $H$, $\psi\colon V(G)\to V(H)$, is a Locally Bijective Homomorphism from $...
- 1,623
0
votes
0
answers
115
views
Three questions about the LPA* algorithm
I have a few questions about the LPA* algorithm, I think I know the answers to most of these questions, but I just wanted to be sure.
Here is the pseudocode for reference:
and here is the link to the ...
- 9
0
votes
0
answers
48
views
Breaking ties in A* to produce same path as D*lite
What tie breaking criteria do I need to implement in A* to mimic exactly the same behaviour as D* lite. Ofcourse both algorithms use the same heuristic and cost functions. So basically if I run A* ...
- 9
3
votes
1
answer
93
views
Name for set of vertices that are pairwise within distance two
A 2-stable set (or a distance-two independent set) of a graph $G$ is a set of vertices which are pairwise at a distance greater than 2 in $G$.
Is there a name for a set of vertices which are pairwise ...
- 1,623
2
votes
0
answers
47
views
Existing results around approximation of minimum 2-edge connected Steiner subgraph
Problem $1$: minimum 2-edge connected subgraph
We are given a $2$-edge connected undirected graph $G(V,E)$, and we are asked to find a $\textit{spanning}$ subgraph $H$, with minimum number of edges, ...
- 141
5
votes
1
answer
155
views
Is there a standard axiomatization of graph width parameters?
There are many useful graph properties described as "width parameters" that show up in algorithm analysis (especially for FPT-type algorithms). The most famous example is probably treewidth,...
- 2,313
4
votes
0
answers
75
views
Reducing counting minimal vertex covers to counting minimum cardinality vertex covers
Consider two problems.
Problem 1: Given a graph $G = (V, E)$, find the number of minimum cardinality vertex covers of $G$.
Problem 2: Given a graph $G = (V, E)$, find the number of minimal vertex ...
- 153
0
votes
1
answer
90
views
On structure of graphs with average degree equal to maximum average degree
For a simple graph $G$, the $\text{average-degree}(G)=|E(G)|/|V(G)|$ and
the maximum average degree $\text{mad}(G)=\max\{\text{average-degree}(H)\colon H \text{ is a subgraph of } G\}$.
If $\text{...
- 1,623
2
votes
0
answers
55
views
Is there a poly-time algorithm to compute the drawing of a simple graph (need not be planar) in a 2D-plane such that any two edges cross at most once?
Does there exists a ploynomial time algorithm to embed a simple graph(need not be planar) in a plane satisfying the following conditions?
No edge touches vertices other than its end vertices.
At any ...
1
vote
0
answers
168
views
Easier famility of graphs for MAXCUT [closed]
I would like to know if there are particular family of graphs for which the Goemans-Williamson MAXCUT Approximation Algorithm renders higher than 0.878 approximation ratio. TIA
- 697
1
vote
0
answers
107
views
Is finding a very large clique NP-hard?
We know that, unless P=NP, for any $\epsilon$ we can't distinguish in polynomial time between the two cases:
There exists a clique of size at least $n^{1-\epsilon}$
All cliques have size at most $n^\...
- 190
0
votes
0
answers
69
views
Is there a term for 'no-turn-back walk' in graph theory?
Let $G$ be a finite undirected graph. A walk in $G$ is a finite sequence $<v_1,e_1,v_2,e_2,\dots,v_{k-1},e_{k-1},v_k>$ where $v_j$'s are vertices in $G$, $e_j$'s are edges in $G$, and $e_j=v_jv_{...
- 1,623
3
votes
0
answers
44
views
Does high connectivity of line graph of $G$ imply high (cyclic) connectivity of $G$?
All graph considered here are finite, simple and undirected.
We know that a graph $G$ is $k$-edge connected if and only if its line graph is $k$-connected (where $k\in\mathbb{N}$). In particular, if $...
- 1,623
2
votes
0
answers
60
views
Summing over weighted paths optimally
Given an edge-weighted directed graph, how do you sum over all weighted paths between A and B while using the smallest number of multiplications?
Is there a name for this problem?
This comes up in ...
- 4,631
3
votes
1
answer
104
views
On cubic planar graphs with face boundaries of length divisible by 4
All graphs considered here are finite, simple and undirected.
Let $\mathscr{G}$ denote the class of cubic plane graphs for which all face boundaries are of length divisible by four. The 3-cube $Q_3$ ...
- 1,623
1
vote
0
answers
40
views
Cycle double covers of cubic graphs using only a few cycles
This is a reference request question. Let $G$ be an arbitrary cubic graph.
Is the problem of finding a cycle double cover $D$ of $G$ with minimum number of cycles in $D$ studied in the literature?
I ...
- 1,623
1
vote
1
answer
182
views
How to show that Color Tiles is NP-Complete
Color Tiles is a puzzle game where color tiles are laid out in a rectangular grid. A tile is visible to an empty grid cells if there is a clear line-of-sight in one of the 4 cardinal directions (You ...
0
votes
0
answers
26
views
How does symmetric difference of b-matchings look like?
It is easy to see that symmetric difference of two matchings are cycles or simple paths. But what about b-matchings? Is there anything known about how they look? Even for restricted cases such as ...
- 109
1
vote
0
answers
29
views
Cycle decompositions of locally linear 4-regular graphs
(Preface)
We consider only finite, simple, undirected graphs here. An orientation of a graph $G$ is obtained by assigning some direction to each edge of $G$.
(Question starts)
A graph is locally ...
- 1,623
0
votes
1
answer
76
views
Is a grid graph a vertex-minor of a complete graph? [closed]
Consider a graph $G$. A graph $H$ is the vertex-minor of the graph $G$ if $H$ can be obtained from $G$ using vertex deletions and local complementations. For more information, look at Definition 2.1 ...
- 153