Let $G$ be a (large) graph and $W$ another (smaller) graph.
$W$ is what I call a *walker*.
Let me use "vertices" and "edges" for $G$ and
"nodes" and "arcs" for $W$.

$W$ has a distinguished node, its *center* $c$.
Say that $W$ is *on a vertex* $v$ of $G$ if
it is *placed* so that node $c$ is on $v$, all
the other nodes of $W$ are on distinct vertices of $G$,
and all arcs of $W$ lie on (distinct) edges of $G$.
So $W$ is a subgraph of $G$, with $c=v$.

Now I'll describe how $W$ "walks."
If $W$ is on $v$, a *step* of $W$
takes $W$ to a placement on a
vertex $u$ of $G$ that is adjacent to $v$ in $G$.
So the center node $c$ of $W$ moves from $v$ to $u$,
and the remainder of $W$ is somehow laid on $G$ so that
it is again a subgraph. Intuitively,
$W$ moves it center, and then redistributes its
"tentacles" around the new center. (One can think
of the walker as a creature that crawls around $G$.)

Finally, say that $W$ can *walk* $G$ if
there is a sequence of steps that moves $W$ so that
its center node $c$ is eventually on every vertex of $G$.
Thus every walkable graph is connected, because
the sequence of placements of $c$ determines a path
that covers all vertices of $G$.

Here is an example. $G$ is an 8-vertex graph,
$W$ a 4-cycle, and the sequence below shows that $W$ can
walk $G$:

I am wondering to what extent the structure of $G$ (beyond its connectedness) is
determined by knowing that $W$ can walk $G$.
For example, $G$ is walkable by a $n$-star $S_n$
(with center at the star hub)
iff
$G$ has minimum degree $n$.
If $W$ is a $n$-cycle $C_n$, then certainly its
girth
is at most $n$,
but I am not sure if walkability by $C_n$
implies any other natural structural constraint on $G$.
I also don't see what walkability by an $n$-path implies.

If anyone knows of a similar concept in the literature, I would appreciate a reference. Walkability is a distant abstraction of a communication network process I was pondering. Thanks!