Questions tagged [graph-minors]
A graph $H$ is called a minor of a graph $G$ if $H$ can be obtained from $G$ by contracting edges, deleting edges, and deleting isolated vertices.
87 questions
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Identifying two non-adjacent vertices and the effect on the Hadwiger number [closed]
Suppose $G$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.
What is an example of a graph $G_0=(V_0, ...
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Two types of criticality
Suppose $G$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number (synonym: connected pseudoachromatic number, cf. [Abrams-Berman 2014, p. 315]) of $G$; that is, the maximum $n\in\mathbb{N}$ ...
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Characterizing SP-DAGs by Forbidden Minors?
So it's well-known that an alternative way to define a series-parallel (undirected graph) is by the forbidden minor $K_4$. Is there a known analog of this definition for directed graphs — ...
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Complete minors in a "redoubled" graph
For any finite, simple, undirected graph $G=(V,E)$ denote by $G_2 = (V_2, E_2)$ the graph, in which $V_2$ and $E_2$ are defined as follows:
$V_2 = \big(V\times\{1\}\big) \cup \big(V\times \{2\}\big)$...
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Complete minors in graphs of bounded diameter
For positive integers $m, d\in \mathbb{N}$ consider the following statement:
$\mathsf{S}(m,d):$ There is $N\in\mathbb{N}$ such that the complete graph $K_m$ on $m$ vertices is a minor of any finite ...
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Is the "surface-minor" ordering of plane graphs a well-quasi-ordering?
A plane graph is a finite simple graph with a fixed embedding into the two-sphere. The embedding induces an embedding on a minors of a plane graph (i.e. a graph obtained by successive removal of ...
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Minors of graphs with infinite chromatic number
Let $G=(V,E)$ be an infinite simple, undirected graph with $\chi(G) \geq \aleph_0$. Is there a minor $M$ of $G$ such that
$M\not\cong G$, and
$\chi(M)=\chi(G)$
?
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Complete minors and minimal degree
Is there for every positive integer $n\in\mathbb{N}$ a finite, simple, undirected graph $G=(V,E)$ with the following property?
$G$ does not have a complete minor with more than $\frac{\delta(G)}{n}$...
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Does anyone know a specific polynomial-time algorithm to detect if a given signed graph contains an odd-K4 as a signed minor?
By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite graphs).
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Is there an established notion of 'signed treewidth' for signed graphs?
By a signed graph, I mean a graph where each edge is designated as either odd or even (as in Guenin's result for weakly bipartite graphs).
It is well-known that for (unsigned, or usual) graphs, ...
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Hadwiger's conjecture in the language of graph homomorphisms
Consider the following statement:
(S): If $G$ is not a complete graph, then there is a minor $M$ of $G$ such that
$M \not \cong G$, and
there is a graph homomorphism $f:G\to M$.
Hadwiger's ...
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Size of forbidden minors for treewidth
For any $k$, the class of graphs of treewidth at most $k$ can be characterized by a finite set of forbidden minors.
For treewidth $1$ and $2$, the set is of size $1$. Then for treewidth $3$, the set ...
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Graph minors, and Kronecker product
Let $X$ and $Y$ be graphs and consider the Kronecker product: $Z = X \otimes Y$. Is it true that if $X$ excludes an $M$-minor, $Z$ excludes an $M \otimes Y$ minor?
I am particularly interested in the ...
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Hadwiger number and minimal degree
Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree of $G$, do we have $\delta(G)\leq\eta(G)$?
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Is there a polynomial-time algorithm to check if a signed graph contains an odd-K5 minor?
I suspect this exists, if anyone has a reference please that would be very helpful.
By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite ...
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An algorithm to compute the number of graphs of size n with a given graph as a minor
I'm looking for any results regarding computing the number of graphs of size $n$ which have a given graph $H$ as a minor. Are there any known algorithms which are more efficient than a brute force ...
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Asymptotics of list size in Robertson-Seymour theorem
A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are ...
user76479
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Is there a version of Robertson-Seymour's graph minor theorem known to apply to signed graphs?
Here a signed graph is one where each edge is signed either odd or even. A cycle is odd or even according to the sum of the signs of its edges. For a given signed graph, a resigning may be performed ...
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A separation property of graphs of bounded tree-width
The following separation property of trees is well-known and in fact easy to prove (see e.g. the paper "Covering a hypergraph of subgraphs" by Noga Alon, Lemma 2.2)
Let $T$ be a tree and $r, m$ non-...
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What is the relation between Hadwiger number and Treewidth?
Is there any general relation between Hadwiger number and Treewidth of a graph? Intuitively I think Hadwiger number is greater than or equal to Treewidth, but I couldn't prove it.
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What is the relation between size of maximum clique and branchwidth?
Let $bw(G)$ be the branchwidth of graph $G$ and $\omega(G)$ be the size of maximum clique in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
Intuition: Assume (in reverse of ...
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A claim from "Graph minors - a survey" by Robertson and Seymour
Can someone give me a proof sketch for this:
Let $\mathscr{P}_n$ be the set of all graphs which do not contain a path on $n$ vertices as a subgraph. Define the type of a graph inductively as: the type ...
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Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
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Number of edges in linklessly embeddable graphs
What is the maximum number of edges of an $n$-vertex linklessly embeddable graph?
A more general question is the following. What is the maximum number of edges of an $n$-vertex graph with Colin de ...
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Contracting a planar graph to a (multiply-edged)-tree
Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions* in order to obtain a graph $T$ which has no loops, and if one ignores parallel edges, $T$ is a ...
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A variant of Kruskal's theorem
For $X$ and $Y$ finite sequences of finite trees, let us say that $X$ is everywhere contained in $Y$ ($X\subseteq_{ec}Y$) iff, for every $y\in Y$, there is some $x\in X$ such that $x$ is a minor of $y$...
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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 $G$...
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Do graphs with large number of cycles always contain large necklace minor?
Let "$k$-necklace" denote the (multi)graph obtained from a cycle of length $k$ by duplicating every edge.
Note that the number of cycles in $k$-necklace is at least $2^k.$
Question : Suppose a ...
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Big binary tree as an induced subgraph
I believe this is true:
Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an
induced subgraph which is a subdivision of a large binary tree or the line ...
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Upper bound on size of obstruction set for wye-delta-wye reducible graphs
A graph is $Y \Delta Y$-reducible if it can be reduced to an empty graph by the following operations:
$Y \leftrightarrow\Delta$ transforms;
Replacing multiple edges with single edges (parallel ...
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Graph minor check
Are there any good algebraic/algorithmic tools available to check if a given graph $H$ is a minor of $G$ from the adjacency matrix of $G$?
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Planar minor graphs
The theorem of Robertson-Seymour about graph minors says that there exists no infinite family of graphs such that none of them is a minor of another one.
Apparently, it came as a generalization of ...
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Minor-closed classes of graphs with large numbers of excluded minors
Robertson and Seymour tell us that any minor-closed family of graphs has a finite collection of excluded minors.
Standard examples include planar graphs with two excluded minors ($K_5$ and $K_{3,3}$) ...
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About Hadwiger's conjecture
Reading the Wikipedia article about Hadwiger's conjecture, I found this open problem really interesting. In this article it is written that
"in a minimal $k$-coloring of any graph $G$, contracting ...
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Can a problem be simultaneously polynomial time and undecidable?
The Robertson-Seymour theorem on graph minors leads to some interesting conundrums.
The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
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Obstructions for embedding a graph on a surface of genus g
Kuratowski's theorem tells us the complete graph $K_5$ and the bipartite graph $K_{3,3}$ are the only obstructions to a graph being planar, ie embeddable in the plane with no edge-crossings.
Is the ...
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Ref request: A graph G contains H as a minor iff it contains one of finitely many graphs as a topological minor
For definitions of graph minors and topological minors, see wikipedia's article on graph minors.
Theorem: For every graph H, there is a finite set of graphs, say S(H), such that G contains H as a ...
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