We say that a simple, undirected graph $G = (\omega, E)$ on the vertex set $\omega$ is tightly knit if there is a positive integer $n>2$ such that for all $v,w\in \omega$ there is a cycle $C$ of length $\leq n$ in $G$ such that $v,w\in C$.
Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic tightly knit graphs on $\omega$?
Consider the graph $G$ with vertices $ \left\{ (x,A) |x ∈\mathbb{N} \right\} \cup \left\{ (x,B) |x ∈\mathbb{N} \right\} \cup \left\{ (x,y) |x ∈\mathbb{N}, y ∈\mathbb{N},y \leq x \right\}$ with two vertices $(w,x)$ and $(y,z)$ connected iff $w=y$ or both $x$ and $z$ are letters.
For each $n\geq 2$, one can decide whether to remove the edge $(n,n-1)$ or not; this gives $2^{\aleph_0}$ graphs.
