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.
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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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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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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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Increasing the Hadwiger number by identifying non-adjacent points
This is a specialization of a more general, still unanswered question.
Suppose $G$ is a finite, simple graph. Let $h(G)$ denote the Hadwiger number, that is, the maximum $n\in\mathbb{N}$ such that $...
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Do graphs with large number of paths contain large chain minor?
Definition: A "$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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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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Complete minors of the grid graphs $\mathbb{Z}^n$
Let $n>1$ be an integer. We say that two points $(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{Z}^n$ are a member of the edge set $E_n$ if and only if $$\sum_{i=1}^n|x_i-y_i| = 1.$$
Question. Given ...
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Large complete minors of $\mathbb{Z}^\omega$
Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x_i - y_i|=1$ and $ x_k = y_k$ for all $k\in \omega\setminus\{i\}$.
$K_\omega$, the ...
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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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