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Zoo Botanica Aventurica 23.pdf

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The introductory information (and a bit about the artwork) are in the first page. The. a pdf. Amazon.com: Book: Zoo-Botanica Aventurica: DSA-Regelerganzung by Adam.Q:

Does every graph is closed under taking minors have bounded degree?

[Question from the book Graphs and Algorithms by Daniel Král]
Let $G$ be a graph. If there exists $k \in \mathbb{N}$ such that $G$ contains no subgraph isomorphic to the complete bipartite graph $K_{k,k}$, then $G$ is called $k$-closed.
Does every $k$-closed graph $G$ has bounded degree? i.e. there exists $d\in\mathbb{N}$ such that every vertex of $G$ is of degree at most $d$?
Also, I know that every cacti-free graph is $d$-closed for every $d\in\mathbb{N}$.

A:

Yes.
Let $C=v_1,v_2,\ldots,v_k,v_k,v_1$ be a cycle in your graph. Let $A$ be the set of all vertices that are not in this cycle. Since $C$ is closed under taking minors, you can find a subgraph of the form $K_{2,k}$ in $G$. Since $A$ is disjoint from this $K_{2,k}$, you can find a path from $v_1$ to $v_k$ whose internal vertices do not intersect $A$. It is easy to construct such a path, and by repeating this argument on the remaining $k-2$ vertices of $A$, you will get a path that has $d$ vertices in total.

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