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  • $\begingroup$ Quite remarkable that all vertices can be shy, thanks for this example! $\endgroup$ Commented Mar 8, 2024 at 21:02
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    $\begingroup$ Concretely, let the vertices be finite sequences $(a_1,a_2,\dots,a_n)$ with $a_i\in\{1,2,\dots,i\}$ and make $(a_1,\dots,a_n)$ adjacent to $(a_1,\dots,a_n,a_{n+1})$. $\endgroup$ Commented Mar 12, 2024 at 2:45
  • $\begingroup$ @bof ah, a graded tree with every vertex on the $n$-the level having $n$ children $\endgroup$ Commented Mar 12, 2024 at 5:28