  Structural graph parameters, such as treewidth, pathwidth, and clique-width,
are a central topic of study in parameterized complexity. A main aim of
research in this area is to understand the "price of generality" of these
widths: as we transition from more restrictive to more general notions, which
are the problems that see their complexity status deteriorate from
fixed-parameter tractable to intractable? This type of question is by now very
well-studied, but, somewhat strikingly, the algorithmic frontier between the
two (arguably) most central width notions, treewidth and pathwidth, is still
not understood: currently, no natural graph problem is known to be W-hard for
one but FPT for the other. Indeed, a surprising development of the last few
years has been the observation that for many of the most paradigmatic problems,
their complexities for the two parameters actually coincide exactly, despite
the fact that treewidth is a much more general parameter. It would thus appear
that the extra generality of treewidth over pathwidth often comes "for free".
  Our main contribution in this paper is to uncover the first natural example
where this generality comes with a high price. We consider Grundy Coloring, a
variation of coloring where one seeks to calculate the worst possible coloring
that could be assigned to a graph by a greedy First-Fit algorithm. We show that
this well-studied problem is FPT parameterized by pathwidth; however, it
becomes significantly harder (W[1]-hard) when parameterized by treewidth.
Furthermore, we show that Grundy Coloring makes a second complexity jump for
more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy
Coloring nicely captures the complexity trade-offs between the three most
well-studied parameters. Completing the picture, we show that Grundy Coloring
is FPT parameterized by modular-width.
