  Trigraph list homomorphism problems (also known as list matrix partition
problems) have generated recent interest, partly because there are concrete
problems that are not known to be polynomial time solvable or NP-complete. Thus
while digraph list homomorphism problems enjoy dichotomy (each problem is
NP-complete or polynomial time solvable), such dichotomy is not necessarily
expected for trigraph list homomorphism problems. However, in this paper, we
identify a large class of trigraphs for which list homomorphism problems do
exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and,
in particular, include all trigraphs whose underlying graphs are trees. In
fact, we show that for these tree-like trigraphs, the trigraph list
homomorphism problem is polynomially equivalent to a related digraph list
homomorphism problem. We also describe a few examples illustrating that our
conditions defining tree-like trigraphs are not unnatural, as relaxing them may
lead to harder problems.
