Some Results on More Flexible Versions of Graph Motif

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.