Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,295
questions
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What is the complexity of this weighted b-edge matching problem?
I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem:
Input: An undirected multigraph $G = (V, E)$ without
loops, an edge partition $(E_1,...
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5answers
2k views
Finding outer face in plane graph (embedded planar graph)
I have a planar graph, for which I have computed a combinatorial embedding and coordinates in the plane. So all my arcs are now oriented in the plane, respective to their end vertices.
Computing a ...
1
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1answer
59 views
Reference request: Depth- (or Breadth-) first search with hints?
Consider the standard s-t reachability problem of finding a path between nodes $s$ and $t$ in a directed graph $G$. A DFS or BFS could solve it.
Would it be possible to pre-process the graph and ...
-4
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1answer
33 views
Distinguish Graph from Tree using Adjacency Matrix
Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle).
For example, given the adjacency matrix:
...
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0answers
23 views
Best known bounds on feedback arcset in high-girth directed graphs?
I asked this question over at MathOverflow, but thinking about it a little more I think it is a more natural fit here.
Let $G$ be a directed graph with $n$ vertices and $m$ edges such that every ...
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0answers
50 views
Describing graphs in terms of subgraphs
I want to create a graph description language that's based on defining and using parameterized subgraphs as "building blocks", specifying exactly how these blocks connect. My use case is describing ...
2
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0answers
72 views
Variation of edge-disjoint spanning trees
In a directed graph, I want to find 2 edge-disjoint spanning trees (arborescence), with the extra restrictions that edges in the 1st tree are not forward arcs in the 2nd tree. Are there existing ...
1
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1answer
126 views
Fast Computation of First k Eigenvectors of Graph Laplacian
I'm looking for references for the following; Given the unnormalized Laplacian matrix of a graph $L = D - A, ~ L \in \mathbb{R}^{n\times n}$
Algorithm(s) to efficiently estimate the first $k$ (...
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0answers
160 views
Sublinear time path existence
Consider a graph $G = (V,E)$ with $N$ edges. Consider two vertices $u_1, v_1 \in V$. We wish to find whether there exists a path of length $4$ between these two vertices or not. This is easy to do in $...
1
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1answer
48 views
When can partial spectral sparsifiers be combined?
A few important papers about spectral sparsifiers and friends contain a technical idea that involves building many different sparsifiers that each "partially" solve the problem, and then combining ...
0
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2answers
106 views
List of NP-Complete graph problems/ properties?
Is there a good source to find various decision problems on graph and networks? For a project I'm doing it'd be useful to be able to look at lots of different problems. Is there a good source for ...
1
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1answer
53 views
Reachability Query for Tree
What is the best complexity for reachability queries on trees so far please? There is no constraint on the directions of the edges in the tree. According to Mikkel Thorup, there is an oracle of size $...
0
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1answer
120 views
Problems that are NP-hard to approximate even when the input graph is regular
Are there graph problems which are NP-hard to approximate even when the input graph is regular?
For instance, are there optimization problems that are NP-hard to approximate within $O(n^\epsilon)$ ...
0
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1answer
87 views
Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?
We conjecture that Hamiltonian cycle is fixed parameter tractable with parameter clique cover, given $k$-clique cover.
Let $G$ be connected simple graph.
$k$-clique cover of graph $G$ is partition ...
5
votes
1answer
122 views
Known property? Maximum radius of connected induced subgraph
I was wondering if the following graph property has a name and has been researched:
Consider any connected induced subgraph $H \subseteq G$.
Then $r(G)$ denotes the maximum radius of any such $H$.
I ...
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1answer
135 views
Minimum vertex cover and odd cycles
Suppose we have a graph G without odd cycles. Consider the minimum vertex cover problem of G formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, ...
3
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0answers
88 views
Isomorphic subforest problem
I recently read that the following problem is NP-Complete:
Given a tree $T$, and a forest $F$, is there a subgraph of $T$ isomorphic to $F$?
The reference provide was to the textbook “Computers and ...
12
votes
1answer
865 views
Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$
I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
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0answers
85 views
How to approach the “traveling salesman problem” with cost changing every time salesman reaches a new city
Let's say instead of finding the shortest path we have to maximize the profit in a year of the salesman under the following constraints.
Salesman can go to a different city only on weekends, all ...
4
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1answer
120 views
Hardness of finding if a vertex lies on a simple directed path between two vertices
Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$?
I found a couple of hardness ...
0
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1answer
58 views
Reduction graph to planar bounded treewidth and bounded diameter graph
We got reduction graph to planar bounded treewidth graph,
but this is unlikely to be true.
Let $H$, the planarizing gadget, be planar graph with four
distinguished vertices $u,u',v,v'$ on the outer ...
5
votes
1answer
177 views
Is this a known problem, and is it #P-complete?
Let $G=(V,E)$ be an undirected graph. What I call a selection function of $G$ is a partial function $f:V \to E$ such that for every node $v$, if $f(v)$ is defined then it is one of the adjacent edges ...
2
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1answer
49 views
Maximum subgraph problem with unknown complexity
Let $Q$ be a polynomial time decidable graph property. In a graph let us call a subgraph $S$ a $Q$-subgraph, if $S$ has the property $Q$. Consider the following optimization problem:
Maximum $Q$-...
1
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1answer
50 views
Existence of graphs of every order related to Barnette’s conjecture
Consider the class C3CBP of $3$-connected cubic bipartite planar graphs. They form the class on which the (in)famous Barnette’s conjecture is based. My interest in C3CBP graphs is somewhat orthogonal ...
6
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2answers
144 views
$NP$-Completeness of $\epsilon$-balanced graph partitioning for fixed $\epsilon$
Consider this graph partitioning problem: Let $G = (V, E)$ be a simple undirected graph and $0 \leq \epsilon \leq 1, M \geq 0$ be constants. Are there disjoint subsets $V_1, V_2$ with $V = V_1 \cup ...
9
votes
2answers
113 views
Complexity of finding an edge set yielding specified vertex degrees
I'm trying to figure out if the following two problems are known in general to be in P or NP-complete:
Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'...
12
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0answers
132 views
NP-Hardness of 4-cycle packing problem in complete bipartite digraph?
A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
7
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0answers
135 views
Algebraic methods for testing planarity
Mac Lane's planarity criterion states that a graph is planar if and only if it has a cycle basis such that each edge is contained in at most two cycles, we call such a basis a 2-basis. A 2-basis of a ...
6
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1answer
199 views
Complexity of finding the largest induced subgraph with all even degrees
What is the complexity of the following problem?
Instance: Simple, undirected graph $G$, and a positive integer $k$.
Question: Does $G$ have an induced subgraph on at least $k$ vertices, such that ...
13
votes
1answer
4k views
What is the correct definition of $k$-tree?
As the title says, what is the correct definition of $k$-tree? There are several papers that talk about $k$-trees and partial $k$-trees as alternative definitions for graphs with bounded treewidth, ...
5
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0answers
138 views
How much does treewidth changes after removal of a path?
Let $G$ be a graph such that $\mathrm{tw}(G)=t$. Let $t' = \min\limits_{u,v \in V(G)} \max\limits_{P \text{ is a path from } u \text{ to } v} \mathrm{tw}(G - P)$. Then how small $t'$ can be?
My ...
7
votes
2answers
178 views
Computing the edge orbits of a graph (and discussing definitions)
A (vertex) automorphism in a graph $G=(V,E)$ is a permutation $\sigma$ of the vertices that preserves adjacency, namely $\sigma(u) \sigma(v) \in E$ if and only if $uv \in E$. The automorphisms of a ...
7
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0answers
188 views
Lighting up all elements of a poset by toggling upsets
I consider the following game on a finite poset $(P, <)$. At each point of the game, I have a set of elements $S$ of the poset which are "on", and all others are "off". Initially $S = \emptyset$. ...
8
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2answers
473 views
Dijkstra parallelization
I'd like to know what is the best method to parallelize the Dijkstra algorithm.
Thanks.
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1answer
102 views
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8answers
3k views
NP-hard problems on paths
everybody knows there exist many decision problems which are NP-hard on general graphs, but I'm interested in problems that are even NP-hard when the underlying graph is a path. So, can you help me to ...
1
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1answer
123 views
Breaking cycles in network graph by adding nodes and rerouting edges
I have a quite "common" need : making a directed graph (with one or several cycles) a directed acyclic graph (DAG).
But the way I want to achieve it is, I guess, way more specific : I would like to ...
2
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0answers
91 views
How hard is it to determine ex(n,G)?
Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
24
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5answers
10k views
Vertex Cover applications in the real world
What applications does the Vertex Cover Problem have in the real world?
Which industry or research projects use actually implemented software that is based on theoretical results for the Vertex Cover ...
0
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1answer
172 views
Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges
A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
1
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1answer
124 views
Finding simple fixed length paths in directed graphs
Is there an efficient algorithm to enumerate unique simple fixed-length paths (of size $k$) in directed graphs? What would be its time complexity?
9
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1answer
121 views
The source of the modular decomposition graph
When introducing graph modular decomposition, most authors use the 11-vertex graph, which I copy from wikipedia.
The question is who is (are) the original designer of it. (I'm not asking who drew ...
25
votes
1answer
693 views
Regularity Lemma for Sparse Graphs
Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets ...
0
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0answers
102 views
Network Reliability Problem
Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
0
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1answer
85 views
Dividing a complete graph into two cliques with maximal sum of edge weights
Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having $\...
9
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2answers
279 views
How long does it take to find a short cycle in a random graph?
Let $G \sim G(n, n^{-1/2})$ be a random graph on $\approx n^{3/2}$ edges. With very high probability, $G$ has many $4$-cycles. Our goal is to output any one of these $4$-cycles as quickly as ...
10
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2answers
194 views
“Relatives” of the shortest path problem
Consider a connected undirected graph with non-negative edge weights, and two distinguished vertices $s,t$. Below are some path problems that are all of the following form: find an $s-t$ path, such ...
3
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0answers
134 views
Enumerating Minimal (a,b) vertex separators in a DAG
A vertex subset $S \subseteq V$ is an $(a,b)$ separator for nonadjacent vertices $a$ and $b$ if the removal of $S$ from a graph $G$ separates $a$ and $b$ into distinct connected components.
$S$ is a ...
2
votes
1answer
114 views
Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free
What is the relationship between graphs which have $O(\log n)$ treewidth and $\mathcal{H}$-minor-free graphs? Are graphs which have $O(\log n)$ treewidth $\mathcal{H}$-minor-free? I know that graphs ...
0
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0answers
64 views
Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color
One can consider the following generalization of the $k$-Coloring problem:
Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
