  Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x(xy),
s_y(xy)} where s_u(uv) is the sum of all weights of vertices that are closer to
u than to v in T. Given a set of weighted vertices V and a multiset of splits
S, we consider the problem of constructing a tree on V whose splits correspond
to S. The problem is known to be NP-complete, even when all vertices have unit
weight and the maximum vertex degree of T is required to be no more than 4. We
show that the problem is strongly NP-complete when T is required to be a path,
the problem is NP-complete when all vertices have unit weight and the maximum
degree of T is required to be no more than 3, and it remains NP-complete when
all vertices have unit weight and T is required to be a caterpillar with
unbounded hair length and maximum degree at most 3. We also design polynomial
time algorithms for the variant where T is required to be a path and the number
of distinct vertex weights is constant, and the variant where all vertices have
unit weight and T has a constant number of leaves. The latter algorithm is not
only polynomial when the number of leaves, k, is a constant, but also
fixed-parameter tractable when parameterized by k. Finally, we shortly discuss
the problem when the vertex weights are not given but can be freely chosen by
an algorithm.
  The considered problem is related to building libraries of chemical compounds
used for drug design and discovery. In these inverse problems, the goal is to
generate chemical compounds having desired structural properties, as there is a
strong correlation between structural properties, such as the Wiener index,
which is closely connected to the considered problem, and biological activity.

Data Structures and Algorithms;Discrete Mathematics