  Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used
to color $V$. The Graph Motif (or GM) problem takes as input $G$, a multiset
$M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$
such that (i) $G[S]$ is connected and (ii) the multiset of colors obtained from
$S$ equals $M$. The Colorful Graph Motif (or CGM) problem is the special case
of GM in which $M$ is a set, and the List-Colored Graph Motif (or LGM) problem
is the extension of GM in which each vertex $v$ of $V$ may choose its color
from a list $\mathcal{L}(v)\subseteq C$ of colors.
  We study the three problems GM, CGM, and LGM, parameterized by the dual
parameter $\ell:=|V|-|M|$. For general graphs, we show that, assuming the
strong exponential time hypothesis, CGM has no $(2-\epsilon)^\ell\cdot
|V|^{\mathcal{O}(1)}$-time algorithm, which implies that a previous algorithm,
running in $\mathcal{O}(2^\ell\cdot |E|)$ time is optimal [Betzler et al.,
IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to $\ell$
even if we restrict ourselves to lists of at most two colors. If we constrain
the input graph to be a tree, then we show that GM can be solved in
$\mathcal{O}(3^\ell\cdot |V|)$ time but admits no polynomial-size problem
kernel, while CGM can be solved in $\mathcal{O}(\sqrt{2}^{\ell} + |V|)$ time
and admits a polynomial-size problem kernel.
