  We study the NP-hard Fair Connected Districting problem recently proposed by
Stoica et al. [AAMAS 2020]: Partition a vertex-colored graph into k connected
components (subsequently referred to as districts) so that in every district
the most frequent color occurs at most a given number of times more often than
the second most frequent color. Fair Connected Districting is motivated by
various real-world scenarios where agents of different types, which are
one-to-one represented by nodes in a network, have to be partitioned into
disjoint districts. Herein, one strives for "fair districts" without any type
being in a dominating majority in any of the districts. This is to e.g. prevent
segregation or political domination of some political party. We conduct a
fine-grained analysis of the (parameterized) computational complexity of Fair
Connected Districting. In particular, we prove that it is polynomial-time
solvable on paths, cycles, stars, and caterpillars, but already becomes NP-hard
on trees. Motivated by the latter negative result, we perform a parameterized
complexity analysis with respect to various graph parameters, including
treewidth, and problem-specific parameters, including, the numbers of colors
and districts. We obtain a rich and diverse, close to complete picture of the
corresponding parameterized complexity landscape (that is, a classification
along the complexity classes FPT, XP, W[1]-hard, and para-NP-hard).
