All Questions
Tagged with graph-minor graph-theory
26 questions
2
votes
1
answer
154
views
Embedding degree-3 planar graphs as topological minors in wall graphs in polynomial time
For a proof, I need the fact that we can efficiently embed an input planar graph into a representative of a specific family of high-treewidth graphs. Specifically, I need an embedding as a topological ...
- 9,838
9
votes
1
answer
552
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 $\mathcal G$ ...
- 14.2k
13
votes
1
answer
1k
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?
5
votes
0
answers
92
views
Complexity of bounded degree full contraction
This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows:
Instance: A graph $G$ and two integers $d$ and $k$.
Question: Is there a ...
- 848
4
votes
1
answer
193
views
Properties of toroidal graph
I am interested in work pertaining to graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below.
Since toroidal graphs can be recognized in $P$ , ...
- 1,016
4
votes
1
answer
292
views
Minor and subdivision
It is a well known fact that if $H$ is a graph of maximum degree 3, then $H$ is a minor of a graph $G$ if and only if $H$ is a topological minor of $G$.
Moreover, a graph $G$ has one of $K_{3,3}$ or $...
- 1,016
6
votes
0
answers
157
views
Book/ Monograph on graph minor theory [Reference request]
I want to learn graph minor theory. Now i have read the very basic things and the overview from the book of R.Diestel but proceeding further is getting difficult. Currently, I am also following the ...
- 1,016
4
votes
1
answer
134
views
Hadwiger number under matching contraction
Given a graph $G$ with Hadwiger number $h(G)$ and a matching $M$ of $G$. Let $G/M$ be the simple graph obtained by contracting $M$. I am looking for a lower bound on the Hadwiger number of $G/M$ as a ...
- 305
9
votes
2
answers
315
views
Something-Treewidth Property
Let $s$ be a graph parameter (ex. diameter, domination number, etc)
A family $\mathcal{F}$ of graphs has the $s$-treewidth property if there is a function $f$ such that for any graph
$G\in \mathcal{...
- 696
19
votes
1
answer
504
views
Minor closed properties that are explicitly MSO expressible
Below, MSO denotes the monadic second order logic of graphs with vertex-set and edge-set quantifications.
Let $\mathcal{F}$ be a minor closed family of graphs. It follows from Robertson and Seymour'...
14
votes
1
answer
397
views
Does treewidth $k$ imply the existence of a $K_{1,k}$ minor?
Let $k$ be fixed, and let $G$ be a (connected) graph. If I'm not mistaken, it follows from the work of Bodlaender [1, Theorem 3.11] that if the treewidth of $G$ is roughly at least $2k^3$, then $G$ ...
- 3,170
5
votes
1
answer
346
views
Sparser Bipartite graphs?
Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members.
Let $\mathcal{C}$ be a ...
- 2,327
9
votes
2
answers
309
views
Understanding graph minor theorem
This question is two-fold, and is mainly reference-oriented:
Is there somewhere where the main intuitions for proving graph minor theorem are given, without going too much into the details? I know ...
- 9,018
8
votes
2
answers
207
views
Do graphs with large number of paths contain large chain minor?
Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
Question: Let $G$...
- 143
13
votes
2
answers
280
views
Complexity of computing a densest minor
Consider the following problem.
Input: An undirected graph $G=(V,E)$.
Output: A graph $H$ which is a minor of $G$ with the highest edge density among all minors of $G$, i.e., with the highest ratio $|...
- 1,116
4
votes
1
answer
193
views
H-induced Containment problem
In the paper "On Graph Contractions and Induced Minors" by Pim van't Hof et al. they showed that this problem is fi xed parameter tractable in |VH| if G belongs to any non-trivial minor-closed graph ...
- 1,016
7
votes
1
answer
354
views
Complexity of finding large grid minors
What is the complexity of finding the largest $k\times k$ grid graph that is a minor of a given graph $G$? It is FPT in $k$, and it seems likely to be NP-hard (or NP-complete in a decision version ...
- 51.2k
8
votes
2
answers
586
views
Grid minor in digraphs
Thor Johnson, et al, in their paper: Directed Tree Width, introduced a definition for directed grid $J_k$, and they conjectured:
$(5.1)$ For every integer $k$ there exists an integer $N$ such that ...
- 3,450
11
votes
1
answer
444
views
MSO properties, planar graphs and minor-free graphs
Courcelle's theorem states that every graph property definable in monadic second-order logic can be decided in linear time on graphs of bounded treewidth. This is one of the most well-known ...
- 10.6k
6
votes
0
answers
335
views
Citation request: Complexity of determining if a graph exists with a minor but no subgraph in set
I'm looking for a reference of the complexity of the following problem.
Let our input $C$ be a finite set of graphs. Is there a graph $G$ such that:
$G$ has a minor in $C$
$G$ has no subgraph in $C$?...
- 343
15
votes
1
answer
518
views
Decomposing graphs of genus one
Planar graphs are $K_{3,3}$-free. Such graphs can be decomposed into tri-connected components, which are known to be either planar or $K_5$ components.
Is there such a "nice" decomposition of ...
- 10.6k
17
votes
2
answers
1k
views
Forbidden minors for bounded treewidth graphs
This question is similar to one of my previous questions. It is known that $K_{t+2}$ is a forbidden minor for graphs of treewidth at most $t$.
Is there a nicely-constructed, parameterized, ...
- 10.6k
17
votes
1
answer
2k
views
Forbidden minors for bounded genus graphs
It is well known that $K_5$ and $K_{3,3}$ are forbidden minors for planar graphs. There are hundreds of forbidden minors for graphs embeddable on a torus. The number of forbidden minors for graphs ...
- 10.6k
12
votes
1
answer
2k
views
Citation showing minors are topological minors for subcubic graphs
If $G$ is a graph with maximum degree 3 and is a minor of $H$, then $G$ is a topological minor of $H$.
Wikipedia cites this result from Diestel's "Graph Theory". It's listed as Prop 1.7.4 in the ...
- 343
3
votes
0
answers
206
views
Name for relationship where one graph is a minor usually implies another is?
Let $G$ and $H$ be graphs with the following relationship: for some $k$, after you perform at least $k$ arbitrary subdivisions of the edges of $G$ (or the edges produced through subdivision), $H$ must ...
- 343
31
votes
5
answers
1k
views
what is easy for minor-excluded graphs?
Approximating number of colorings seems to be easy on minor-excluded graphs using algorithm by Jung/Shah. What are other examples of problems that are hard on general graphs but easy on minor-excluded ...
- 4,731