Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
1,481
questions
2
votes
1
answer
201
views
Does abundance of max cliques make it easy to solve COLORABILITY?
Let $q\geq 3$. We know that $q$-COLORABILITY is an NP-complete problem.
Suppose that $G$ is a graph such that each vertex of $G$ is part of a $q$-clique (i.e. $K_q$). Since we may assume that $G$ does ...
- 1,603
1
vote
0
answers
45
views
Approximate solution for maximum coverage problem with choice constraint
Suppose a sequence of sets $S_1,S_2,...,S_i$ where each set contains sets of elements. That is, each set $S$ contains many sets $a_1,a_2,...,a_{|S|}$. We are given an integer $k$ and we assume that $\...
0
votes
0
answers
22
views
Listing colourful Eulerian orientations in poly. time
An orientation $\overrightarrow{G}$ of an undirected graph $G$ is a directed graph obtained by assigning some direction on every edge of $G$.
An orientation $\overrightarrow{G}$ is said to be an ...
- 1,603
0
votes
0
answers
44
views
Is there a primal-dual algorithm for the Tree Augmentation Problem or the Cactus Augmentation Problem?
The TAP problem and the CacAP problem can be seen as covering problems for the minimum cuts of a graph.
It seems like these problems would fall under the framework of network design problems (...
- 557
2
votes
0
answers
68
views
What is the expected treewidth of a large-treewidth graph intersected with Erdos-Renyi graph?
Suppose we have a graph $G$ with treewidth $t$. Let $p \in (0,1)$ be a constant. Then let's independently remove each edge from $G$ with probability $p$. What is the expected treewidth of the ...
- 309
3
votes
1
answer
191
views
Are there an algorithm that find Minimum spanning tree in $O(n^2\log\log^*n)$?
Given completed metric weighted graph $G=(V,E)$ that have $n$ vertices. Are there an algorithm that find MST of $G$ in $O(n^2)$?
I read abstract of this paper that mentioned an algorithm with running ...
- 194
4
votes
1
answer
173
views
Does a graph resulting from the union of triangles has a particular name?
I have different simple triangle graphs. As an instance, $G_1=(V_1,E_1)=(\{1,2,3\},\{\{1,2\},\{2,3\},\{3,1\}\})$ and $G_2=(V_2,E_2)=(\{1,4,5\},\{\{1,4\},\{4,5\},\{5,1\}\})$.
The union of both graphs ...
- 69
3
votes
2
answers
192
views
exact path cover for undirected graph
In a Python plotting application,
I have an undirected connected graph, not necessarily simple, that I'd like to cover with paths such that each edge is contained in exactly one path.
The number of ...
- 149
0
votes
0
answers
37
views
Bounded-Frequency Minimum Set Cover Problem
Consider the special case of the minimum set cover problem where each element of the universe occurs in at most 3 sets.
Can this problem be solved in polynomial time?
Is there a nontrivial upper ...
- 101
1
vote
1
answer
194
views
State of the art approximation algorithm for $\text{MAXCUT}$ that does better than Goemans and Williamson
I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia:
The polynomial-time approximation algorithm for Max-Cut with the best
known ...
- 153
0
votes
0
answers
52
views
Minimal lexicographical path on DAG in O(||V| + |E|)
Let's assume, that we have directed asyclic graph and nodes U and V. Every edge of this graph is marked with alphabet letter (alphabet size is fixed). Is there any way to answer, what is the shortest ...
2
votes
1
answer
105
views
Can this special case of Node Weighted Steiner Tree be solved in polynomial time?
Consider the node-weighted steiner problem:
Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$.
Output: a minimum weight subset $S \...
- 557
4
votes
1
answer
367
views
Pagerank in directed *acyclic* graphs (DAG)
I deal with pagerank computations on large directed acyclic graphs (DAG).
I found no reference to work on this specific case, only some work on pagerank in more specific cases, e.g., PageRank of Scale ...
- 683
0
votes
2
answers
310
views
Partition a graph into two clusters
Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized?
...
- 194
5
votes
1
answer
225
views
Computational complexity of Turán-type problems
According to Turán's theorem (with $r=n/2$), any graph $G$ with $n$ vertices and at least $n(n-2)/2$ edges must contain a clique of size $n/2+1$. My question is: how hard is it to find this clique?$^*$...
- 213
2
votes
0
answers
93
views
How typical are odd-H-minor free graphs?
Can anything be said about how typical are odd-H-minor free graphs? (definition of odd-minor-free is in Section 2.2 of notes, page 20 of slides). For instance, for a random graph with $n$ vertices, $...
- 4,631
0
votes
0
answers
22
views
Going from one base packing to another using basis exchanges
Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So ...
- 557
6
votes
0
answers
199
views
Fastest exact algorithm for MAXCUT
Is the algorithm introduced in the following paper still the fastest exact algorithm for general MAXCUT problems? TIA
Ryan Williams, A new algorithm for optimal $2$-constraint satisfaction and its ...
- 997
5
votes
2
answers
260
views
Complexity of "can we get a cycle by stacking directed bipartite graphs?"
Preliminaries
We consider directed bipartite graphs of the form $G = (V,V',E)$, in which the nodes are partitioned into $V = \{1,\ldots,n\}$ and $V'=\{1',\ldots,n'\}$, with $|V|=|V'|=n$, and $E\...
- 103
1
vote
1
answer
101
views
Partition the edges of a bipartite graph into perfect $b$-matchings
Any $r$-regular bipartite graph can be partitioned into $r$ disjoint perfect matchings.
I want to know whether a version of this extends to perfect $b$-matchings.
Suppose we have a bipartite graph $G =...
- 557
0
votes
1
answer
121
views
Max-k-cut with negative edge weights
Let $G=(V,E,w)$ be a graph and for edge $e\in E$, there is associated weight $w_e$.
The max-k-cut wants to partition V into k subsets $P_1,\cdots,P_k$ and maximize $\sum_{1\leq r<s\leq k}\sum_{i\in ...
1
vote
1
answer
50
views
Graph classes where giving a q-clique edge cover makes testing for q-colouring easy
A $q$-clique of a graph is a complete subgraph on $q$ vertices.
A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
- 1,603
1
vote
0
answers
54
views
Prune length distribution of random binary tree
Consider a random binary tree with $N$ leaves. Each node (except the root node) has a degree of exactly three (two children and one parent). No further restriction is placed on the structure of the ...
3
votes
1
answer
101
views
Finding the single-crossing embedding of a single-crossing graph
Is it known how to find a (piecewise) straight-line embedding of a single-crossing graph on the plane with exactly one crossing in polynomial time? We are currently trying to come up with a method for ...
- 175
2
votes
1
answer
160
views
Coloring intersection graph of squares
It is known that the coloring intersection graph of axis-parallel rectangles is NP-Hard.
What about squares and more specific case "unit squares"?
Thanks.
- 21
0
votes
1
answer
51
views
Planar 4-regular vertex-transitive graphs as system of circles
It is known that every planar 4-regular 3-connected graph $G$ admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles such that the vertices of $G$ ...
- 1,603
1
vote
0
answers
145
views
How can I find the PhD thesis of A. V. Kostochka?
I've searched for the doctoral thesis of Alexandr V Kostochka in internet but couldn't find it. Can somebody help me?
I have searched in his publications list (which contains only one article ...
- 1,603
1
vote
1
answer
96
views
Is there a regular bipartite graph where the minimum cuts are trivial?
My question is: Given integers $r$ and $k$, is there an $r$-regular bipartite graph $G = L \cup R$ with $|L| = |R| = k$, which is $r$-edge connected, and such that every minimum cut is trivial?
We can ...
- 557
5
votes
0
answers
73
views
Does every graph of clique-width 3 have a large induced subgraph of clique-width 2?
Is there a constant $\alpha>0$ such that every graph $G$ of clique-width $3$ and order $n$ has an induced subgraph of order at least $\alpha n$ and clique-width at most $2$ (in other words, the ...
- 266
2
votes
1
answer
171
views
Is the difference between the acyclic chromatic number and the star chromatic number unbounded?
Is $\chi_s(G)-\chi_a(G)$ unbounded in general graphs?
I thought $\chi_s(G)-\chi_a(G)$, the difference between the star chromatic number and the acyclic chromatic number, is unbounded for general ...
- 1,603
-2
votes
1
answer
315
views
Find research partner (profession and beginner)
I've 10 years of industrial work, but in my free time, I do research, write papers to conferences, help to teach to my old friend at the university and I even did a Ph.D. full-time program.
Now, I've ...
- 45
1
vote
0
answers
104
views
Has this notion of connectivity in edge-colored graphs been studied?
Consider a simple graph $G$ where each edge is either red or blue. I'm interested in the following notion of connectivity:
Two vertices $u$ and $v$ are said to be connected if there is a path ...
- 266
3
votes
1
answer
173
views
Detect if a graph has a $k$ cycle in space complexity $O((\log k)^d)$ for fixed $d \geq1$
For a graph $G$, I want to test if it contains a cycle of length $k$, for some $k$ much smaller than $|G|$. I am interested in particular in an algorithm with low space complexity. The cycle need not ...
- 133
0
votes
1
answer
54
views
Do such instances always admit a 3D matching?
I want to know whether the following kinds of special instances of the 3D Matching problem are ``yes" instances, i.e., admit a 3D matching.
We are given 3 sets $A,B,C$ containing $m$ elements ...
- 15
3
votes
1
answer
153
views
TSP with "enemy" nodes
I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for.
The story/idea ...
- 407
8
votes
1
answer
180
views
Finding vertex separator such that the induced subgraph has minimal number of edges
My problem is related to edge and vertex cuts with a little twist.
Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
- 183
-1
votes
1
answer
131
views
What are the known classes of undirected graphs such that every graph belonging to that class is guaranteed to have a Hamiltonian Path?
One trivial class of graphs is the class consisting of complete graphs or complete bipartite graphs with equal sized partitions.
I would love to know if more such classes exist.
- 177
-2
votes
1
answer
98
views
upper bound on the total number of fixed-length paths in an acyclic graph [closed]
I was wondering if there is an upper bound on the total number of fixed-length paths (path length from 1 to $n-1$ given $n$ nodes) in an acyclic graph (not directed) of $n$ nodes? If so, can you point ...
- 127
4
votes
1
answer
185
views
Does such a bipartite graph exist?
In the course of my studies on graphs I sometimes use gadgets. I recently came upon a need for a certain bipartite graph with the following properties, and I am wondering if anyone knows if such a ...
- 557
1
vote
0
answers
96
views
Complexity of counting 3-colourings of planar bounded degree graphs
The following are known:
It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1].
For all $d\geq 3$, it is #P-complete to count the number ...
- 1,603
5
votes
0
answers
236
views
How to find the second smallest cut in a graph?
For an undirected graph, how do we find the second smallest $s,t-$cut(s) for some $s,t\in V$? What's the time complexity of this computation? What if we only cared about finding a cut of size $p+1$, ...
0
votes
0
answers
34
views
Listing Eulerian orienations with special properties
An orientation of a simple undirected graph is said to be Eulerian if every vertex has the same number of in-coming edges and out-going edges (i.e., in-degree($v$)=out-degree($v$) for all $v\in V(G)$)....
- 1,603
4
votes
1
answer
230
views
Complexity of relaxed edge colouring
A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
- 1,603
7
votes
0
answers
115
views
Decomposing graph homomorphisms
A homomorphism $h: G\to H$ from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to those of $H$ which preserves edges, that is, if $(x,y)$ is an edge of $G$ then $(h(x),h(y))$ is an ...
- 161
1
vote
0
answers
54
views
Generalizing PageRank for tripartite graphs
Problem
I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...
- 111
3
votes
0
answers
70
views
Number of connected partitions (or labelings) in a grid graph
Let $G$ be a 2D lattice graph (undirected) of size $W\times H$. Each "inner" vertex has $4$ adjacent vertices, whereas "boundary" vertices have $2$ or $3$ adjacent vertices, ...
- 31
0
votes
0
answers
73
views
Prove that this linear relaxation has half-integral extreme points
Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope:
(1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$
(2) For each $e \in E$, $0 \leq x_e \leq 1.$
Here $\delta(S)$...
- 557
1
vote
1
answer
61
views
Weak incidence colouring
Let $G(V,E)$ be a simple undirected graph, let $k\in \mathbb{N}$, and let $I(G)=\{(v,e)\ :\ e\in E, v\in e \}$ (here, $v\in e$ means $v$ is incident on $e$). A $k$-incidence colouring of $G$ is a ...
- 1,603
7
votes
1
answer
148
views
Monotone circuit representations of paths in a graph?
Consider a directed graph $G = (V, E)$ with a source $s \in V$ and sink $t \in V$. From $G$, I can define a monotone Boolean function $\phi_G$ on the set of variables $E$, in the following way: every ...
- 7,455
2
votes
0
answers
99
views
Graph recovery from pairwise-common neighborhoods
Define the common neighborhood of two vertices $u$ and $v$ of a simple undirected graph as the set $N(u,v)=N(u)\cap N(v)$. For a simple bipartite graph $G=(U,V,E)$, define the pairwise-common ...
- 748