  The problems studied in this paper originate from Graph Motif, a problem
introduced in 2006 in the context of biological networks. Informally speaking,
it consists in deciding if a multiset of colors occurs in a connected subgraph
of a vertex-colored graph. Due to the high rate of noise in the biological
data, more flexible definitions of the problem have been outlined. We present
in this paper two inapproximability results for two different optimization
variants of Graph Motif: one where the size of the solution is maximized, the
other when the number of substitutions of colors to obtain the motif from the
solution is minimized. We also study a decision version of Graph Motif where
the connectivity constraint is replaced by the well known notion of graph
modularity. While the problem remains NP-complete, it allows algorithms in FPT
for biologically relevant parameterizations.
