  Let $X$ and $Y$ be two graphs with vertex set $[n]$. Their
friends-and-strangers graph $\mathsf{FS}(X,Y)$ is a graph with vertices
corresponding to elements of the group $S_n$, and two permutations $\sigma$ and
$\sigma'$ are adjacent if they are separated by a transposition $\{a,b\}$ such
that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are
adjacent in $Y$. Specific friends-and-strangers graphs such as
$\mathsf{FS}(\mathsf{Path}_n,Y)$ and $\mathsf{FS}(\mathsf{Cycle}_n,Y)$ have
been researched, and their connected components have been enumerated using
various equivalence relations such as double-flip equivalence. A spider graph
is a collection of path graphs that are all connected to a single center point.
In this paper, we delve deeper into the question of when $\mathsf{FS}(X,Y)$ is
connected when $X$ is a spider and $Y$ is the complement of a spider or a
tadpole.
