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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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 ...
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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 ...
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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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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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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 ...