Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation

In this paper, we show that Bandwidth is hard for the complexity class $W[t]$ for all $t\in \mathbf{N}$, 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 $t\in \mathbf{N}$, for directed acyclic graphs whose underlying undirected graph is a caterpillar.