Dichotomy for tree-structured trigraph list homomorphism problems

Trigraph list homomorphism problems, also known as list matrix partition problems, generalize graph list colouring and digraph list homomorphism problems. While digraph list homomorphism problems enjoy a dichotomy   (each problem is NP-complete or polynomial time solvable), such dichotomy is not necessarily expected for trigraph list homomorphism problems, and in the few cases where dichotomy has been proved, for small trigraphs, the progress has been slow.In this paper, we prove dichotomy for trigraph list homomorphism problems where the underlying graph of the trigraph is a tree. In fact, we show that for these trigraphs the trigraph list homomorphism problem is polynomially equivalent to a related digraph list homomorphism problem. The result can be extended to a larger class of trigraphs, and we illustrate the extension on trigraph cycles.