Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,246
questions
-4
votes
1answer
48 views
Graphs with pairwise distance of log*n
I'm looking for example of graphs where some vertices will have $\log^*n$ distance, where $n$ is the number of nodes in the graph.
Not even sure how it should look like.
Here $\log^* x$ denotes the ...
8
votes
1answer
122 views
Efficient graph isomorphism for similar graph queries
Given the graph G1, G2 and G3, we want to perform isomorphism test F between G1 and G2 as well as G1 and G3. If G2 and G3 are very similar such that G3 is formed by deleting one node and inserting one ...
-2
votes
0answers
82 views
Find all perfect matchings in a general undirected graph
Is there an algorithm which can compute all perfect matchings of a general undirected graph (assuming they exist)?
-3
votes
0answers
58 views
Determine if a graph has exactly 1 cycle using a SAT solver
I have a connected undirected graph whose edges are either enabled or disabled. I want to create a set of clauses that are SAT iff all enabled edges are part of a single loop.
If I assert that each ...
2
votes
1answer
62 views
Computing the existence of a path in a code execution graph
I have a need for an algorithm which I can express as a reachability problem in a graph.
Note that I'd appreciate any advices with respect to better wording this question. Also please tell me if this ...
3
votes
0answers
70 views
Embed graph in $\ell_2$ space so that edge and non-edge distances are separated by a constant factor
Suppose I have an undirected unweighted graph $G = (V,E)$. Is there a way to compute points $x_v \in \mathbb{R}^d$ for each vertex $v \in V$ such that $||x_v - x_u|| = 1$ whenever $(u,v) \in E$ and $ |...
-2
votes
0answers
59 views
Is there an algorithm similar to Stoer–Wagner mincut algorithm?
Suppose I want to find the minimum weight cut in a weighted undirected graph. I see Stoer–Wagner algorithm mincut algorithm does help to find the minimum cut. Is there an algorithm similar to Stoer–...
11
votes
0answers
298 views
A dynamic data structure to list triangles
Consider an undirected graph with $n$ nodes. Is there an efficient data structure that supports the following operations?
Insert an edge into the graph
Delete an edge from the graph
Given a query ...
2
votes
0answers
79 views
Which computational framework lies behind the Chinese “Social Credit System”?
BACKGROUND
The Social Credit System is a data-driven reputation system which draws on several sources to label various entities, namely businesses and individual citizens, with a trustworthiness ...
-1
votes
0answers
46 views
What are constraints for two k-regular graphs to be isomorphic?
As discussed in the answer for this question, it is not necessary that two k-regular graphs are always isomorphic to each other. However, 2-regular graphs are isomorphic.
So my question is if there ...
0
votes
0answers
62 views
Unknown gaps in computation models
I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2.
where T1 is smaller then T2 and it is unknown if there are problems where their ...
0
votes
1answer
71 views
finding maximum weight subgraph
My graph is as follows:
I need to find a maximum weight subgraph.
The problem is as follows:
There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
3
votes
0answers
44 views
Problem of determining if a $4$ connected graph has $k$ Hamiltonian cycles
Definition: Define the $k$-HamiltonianCycles problem as the decision problem that asks if a given graph has at least $k$ distinct Hamiltonian cycles.
Question: Is there some constant $k$ so that the $...
-1
votes
1answer
46 views
Min Cut with Vertices
I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking ...
1
vote
0answers
54 views
Graph automorphism with prescribed values
Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
1
vote
1answer
90 views
How many samples are needed to reconstruct a path?
Consider an input set of vertices $V$ and vertices $s,t\in V$.
The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
0
votes
0answers
19 views
The set of weight functions for which the assignment problem has non-trivial solutions
The standard assignment problem is specified with a square matrix ${\bf W}$ of weights (values, costs):
$$
V_{\cal P} = \sum_i w(i, b(i)) = \sum_{(i, j) \in {\cal P}} w_{ij},
$$
where $\cal P$ is a ...
0
votes
1answer
33 views
Graph path problem [duplicate]
I am trying to solve one graph traversing problem which might be classical to guys who are familiar with the topic. However, I am not. I have directed graph where nodes are cities and plane can fly ...
0
votes
1answer
60 views
Finding a Hamiltonian cycle from perfect matching of a bipartite graph
A disjoint vertex cycle cover of G can be found by a perfect matching on the bipartite graph, H, constructed from the original graph, G, by forming two parts G (L) and its copy G(R) with original ...
-3
votes
1answer
95 views
Finding the maximum no. of people who get along in a group [closed]
Suppose that there are 15 people in a room. Assume that each person gets along with other people in the room (but not everyone). (Note that the "feeling is mutual" between any two people who are ...
1
vote
0answers
75 views
Minimum cut with nonlinear objective function
Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum.
Let us generalize it the following way: let $f$ be a ...
0
votes
1answer
81 views
Densest k subgraph problem for outerplanar graphs?
The densest k subgraph problem aims to find a subgraph $H$ of a graph $G$ with exactly $k$ vertices that maximizes the number of edges $|E(H)|$.
Does anyone know if there exists a polynomial-time ...
3
votes
0answers
83 views
Counting quotient graphs, but not exactly
All graphs considered will be directed graphs $G=(V,E)$, with $E \subseteq V \times V$ (so possibly with self-loops). For $k \in \mathbb{N}_{\geq 1}$, I will write $[k]$ the set $\{1,\ldots,k\}$. A $k$...
0
votes
1answer
166 views
Number of simple paths between two vertices in a DAG
Let $G = (N, A)$ be a connected acyclic digraph (DAG). Furthermore, let $s \in N$ and $t \in N$ be two vertices on this graph, such that $t$ is reachable from $s$.
My problem is: how many simple $s-t$...
5
votes
1answer
149 views
Minimum cost cut with discount - what is the complexity?
Consider an undirected graph $G=(V,E)$ with non-negative edge costs. Given an integer $k$ with $0\leq k\leq |E|$, let us call an edge set $C\subseteq E$ a $k$-discounted cut, if the following hold:
$...
1
vote
1answer
144 views
Does Hadwiger conjecture imply that NP = coNP?
(Disclaimer: I suspect the answer is no, but I fail to see why)
Here is a nice picture by David Epstein (taken from Wikipedia) illustrating Hadwiger's conjecture:
The point is that if in a given ...
0
votes
0answers
38 views
Reference on generalization of plane graph duality between bonds and simple cycles
I've also asked this question on Mathoverflow, but it hasn't gotten an answer after several months: https://mathoverflow.net/questions/316132/reference-on-generalization-of-plane-graph-duality-between-...
2
votes
0answers
108 views
Shortest s-t path when is allowed to ignore k weights
Given an undirected graph $G$ with $n$ vertices and $m$ edges, with non-negative weights on the edges, what's the best algorithm that computes the shortest path from $s$ to $t$, where you are allowed ...
6
votes
1answer
267 views
Naive definition of treewidth
Treewidth has arguably pretty involved definition. Recently I was thinking about a problem and turns out it easy to solve it for graphs with small ``naive treewidth''.
Naive treewidth is defined as ...
4
votes
0answers
137 views
Is there a standard name for this way of modifying graphs?
Let $G = (V, E)$ be an undirected graph. Let me take an edge $\{x, y\}$ (in blue in the drawing) such that $x$ and $y$ have other incident edges. Among the incident edges we choose one edge $e_x = \{...
0
votes
1answer
57 views
Counting sum of parities of cycle covers in cubic, planar, bipartite graphs
Let $G$ be a cubic (i.e. every degree exactly three), planar, bipartite graph. By Hall's theorem its edges can be partitioned into three perfect matchings. Take any such partition $M_0,M_1,M_2$ and ...
3
votes
0answers
103 views
Generating a random connected bipartite graph
A (n, m, k)-bipartite graph is a bipartite graphs with:
independent sets of size $\{n, m\}$
a total of $k \geq n+m-1$ edges
We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
5
votes
0answers
170 views
What's the fastest known algorithm for finding the diameter of a graph?
Given a positively weighted graph what's the fastest algorithm for finding the diameter for that graph?
5
votes
1answer
148 views
Viola's Reduction of 3XOR to listing triangles
Apparently this was due to Pătraşcu, but in this report on the ECCC server, Viola states that 3XOR can be reduced to listing triangles.
Assume that given a graph in adjacency list format, with $m$ ...
1
vote
0answers
124 views
Star seperators to explain computational complexity of algorithms on a class of graphs?
A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
1
vote
1answer
125 views
maximize edges minus vertices in a weighted graph
for a given weighted vertices and edges graph, we want to find the maximum subgraph. the maximum subgraph is made of some vertices and some edges of the given graph which sum of the edges minus sum of ...
6
votes
1answer
132 views
Separating words and graph isomorphism
I wonder if there are any known implications of Babai's recent quasi-polynomial time algorithm for Graph Isomorphism to separating words by DFA's.
In both cases the ultimate goal is to differentiate ...
1
vote
0answers
49 views
Is there an efficient way to reduce a set of graphs under isomorphism?
Suppose we have a set $S$ of (fixed-size) graphs, many of which are isomorphic to each other. How do you find a minimum-size set $M$ of graphs such that every $g\in S$ is isomorphic to some graph in $...
1
vote
1answer
148 views
Is balanced Hamiltonian cycle NP complete on maximal plane graphs?
I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs.
If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
1
vote
1answer
78 views
Pulling a graph across a partition
I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
1
vote
1answer
81 views
Cryptography protocols using graph problem instances
I personally am only aware of basic examples of public key cryptography and I haven't studied cryptography yet. I'm curious if there are circumstances in cryptography where using problem instances ...
2
votes
0answers
129 views
Crime prevention using graph theory and machine learning
I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
9
votes
1answer
304 views
Is there an algorithm that finds the forbidden minors?
The Robertson–Seymour theorem says that any minor-closed family $\mathcal G$ of graphs can be characterized by finitely many forbidden minors.
Is there an algorithm that for an input $\mathcal G$ ...
9
votes
0answers
136 views
Random unbalanced bipartite graphs are good small set expanders
My question is about small set expansion properties of random unbalanced bipartite graphs.
Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
4
votes
1answer
75 views
Complexity of finding Exact Size Cut-Sets in Bipartite Graphs
I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
1
vote
1answer
44 views
Problem property name where an optimal solution in a graph can be used as a solution in any subgraph
Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
5
votes
1answer
114 views
Counting/Enumerating Minimal Edge Covers
A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it.
Questions
Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
12
votes
1answer
839 views
Number of 4 cycles
Let $C_4$ be a cycle with four vertices. For an arbitrary graph $G$ with $n$ vertices and m edges say $m>n\sqrt n$, how many $C_4$s exist? Is there a lower bound for this?
1
vote
0answers
51 views
Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity
I am wondering if this is a topic that has had research done...
If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
8
votes
1answer
188 views
Could chromatic number be easy to calculate when colouring is hard for some graph class?
Similar question was asked before, but there was an error in it so it was left unanswered Graph class with easy chromatic number, but NP-hard coloring
Is there any infinite set of graphs $C$ such as:
...
