# 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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### Reference to a definition of a graph homology

Let $G$ be a graph, and define $C_k$ to be the free abelian group on the following homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
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### Is there is a constant $c$ such that toroidal graphs are minor-$c$-colorable?

A toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross. A minor of graph G is a graph obtained from G by ...
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### What is known about "overlapping" models/minors of graphs

Suppose we are given a graph $G$ from a graph class of sublinear treewidth and connected subgraphs $G_1,G_2,..,G_l$ of $G$ such that each $G_i$ intersects (shares a node with) constant many other $G_j$...
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### Flat or linkless embeddings of graph with fixed projection

The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question ...
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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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### 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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### 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 $n\in\mathbb{N}$ ...
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 — ...
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: $V_2 = \big(V\times\{1\}\big) \cup \big(V\times \{2\}\big)$...