Abstract
The \sc Colorful Motif} problem asks if, given a vertex-colored
graph G, there exists a subset S of vertices of G such
that the graph induced by G on S is connected and contains
every color in the graph exactly once. The problem is motivated
by applications in computational biology and is also
well-studied from the theoretical point of view. In particular,
it is known to be NP-complete even on trees of maximum degree
three~[Fellows et al, ICALP 2007]. In their pioneering paper
that introduced the color-coding technique, Alon et al.~[STOC
1995] show, {\em inter alia}, that the problem is FPT on general
graphs. More recently, Cygan et al.~[WG 2010] showed that {\sc
Colorful Motif} is NP-complete on {\em comb graphs}, a special
subclass of the set of trees of maximum degree three. They also
showed that the problem is not likely to admit polynomial
kernels on forests.
We continue the study of the kernelization
complexity of the {\sc Colorful Motif problem restricted to
simple graph classes. Surprisingly, the infeasibility of
polynomial kernelization persists even when the input is
restricted to comb graphs. We demonstrate this by showing a
simple but novel composition algorithm. Further, we show that
the problem restricted to comb graphs admits polynomially many
polynomial kernels. To our knowledge, there are very few
examples of problems with many polynomial kernels known in the
literature. We also show hardness of polynomial kernelization
for certain variants of the problem on trees; this rules out a
general class of approaches for showing many polynomial kernels
for the problem restricted to trees. Finally, we show that the
problem is unlikely to admit polynomial kernels on another
simple graph class, namely the set of all graphs of diameter
two. As an application of our results, we settle the classical complexity of
\cds{} on graphs of diameter two --- specifically, we show that it is \NPC.
