  In this paper, we show that Bandwidth is hard for the complexity class $W[t]$
for all $t\in {\bf N}$, even for caterpillars with hair length at most three.
As intermediate problem, we introduce the Weighted Path Emulation problem:
given a vertex-weighted path $P_N$ and integer $M$, decide if there exists a
mapping of the vertices of $P_N$ to a path $P_M$, such that adjacent vertices
are mapped to adjacent or equal vertices, and such that the total weight of the
image of a vertex from $P_M$ equals an integer $c$. We show that {\sc Weighted
Path Emulation}, with $c$ as parameter, is hard for $W[t]$ for all $t\in {\bf
N}$, and is strongly NP-complete. We also show that Directed Bandwidth is hard
for $W[t]$ for all $t\in {\bf N}$, for directed acyclic graphs whose underlying
undirected graph is a caterpillar.
