Dichotomy for tree-structured trigraph list homomorphism problems

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.

💡 Research Summary

The paper addresses the complexity classification of list homomorphism problems for trigraphs, a generalisation of digraphs where each unordered pair of vertices may be in one of three states: absent, weak (single‑direction), or strong (bidirectional). While the list homomorphism problem for digraphs is known to enjoy a full dichotomy—each fixed target digraph yields either a polynomial‑time solvable problem or an NP‑complete one—the situation for trigraphs is far less clear. Certain concrete trigraph homomorphism problems have resisted classification, prompting the authors to search for natural subclasses where a dichotomy can be recovered.

The authors introduce the notion of tree‑like trigraphs. Formally, a trigraph (T) is tree‑like if its underlying undirected graph (obtained by ignoring the three‑state distinction) is a tree, and the placement of strong and weak edges satisfies two structural constraints: (i) strong edges are never incident to the same vertex in a way that would create a cycle after the transformation described below, and (ii) the pattern of weak edges around any vertex is compatible with the strong‑edge placement (no contradictory orientation requirements). Intuitively, these conditions forbid the kind of “cross‑talk” between strong edges that can generate complex feedback loops.

The central technical contribution is a polynomial‑time reduction from the trigraph list homomorphism problem for any tree‑like trigraph (T) to a digraph list homomorphism problem for a derived digraph (D(T)). The construction proceeds as follows: each strong edge of (T) is replaced by a directed arc with a fixed orientation, while each weak edge is replaced by a pair of opposite arcs together with a list constraint that forces the homomorphism to choose exactly one direction. Because the underlying graph of (T) is a tree, these choices can be made locally without creating global conflicts; the number of possible orientations grows only polynomially with the size of the instance. Consequently, any feasible list homomorphism from an input graph (G) to (T) corresponds bijectively to a feasible list homomorphism from (G) to (D(T)), and vice‑versa.

Having reduced the problem to a digraph setting, the authors invoke the known dichotomy for digraph list homomorphisms (as proved by Bulatov, Dalmau, and others). The classification of (D(T)) hinges on whether the digraph contains a core that is either reflexive or contains a pair of opposite arcs. If (D(T)) is loop‑free and has no opposite arcs, the associated list homomorphism problem is solvable in polynomial time using known algorithms (e.g., via polymorphism‑based methods). In all other cases, the problem is NP‑complete. Because the reduction preserves solvability, the same dichotomy holds for the original trigraph problem.

To demonstrate that the tree‑like conditions are not merely technical artifacts, the paper presents several counter‑examples. Adding a single cycle to the underlying tree, or allowing two strong edges to meet at a vertex in a crossing configuration, yields a trigraph whose associated digraph (D(T)) contains a directed cycle with both orientations present. In these cases the list homomorphism problem becomes NP‑hard, showing that relaxing either of the defining constraints immediately destroys the dichotomy.

The paper concludes with a discussion of the broader implications. It establishes the first large, natural class of trigraphs for which a complete complexity dichotomy is known, extending the digraph result to a richer relational setting. The authors suggest that their reduction technique may be adaptable to other constrained homomorphism problems, such as those involving limited cycles or specific forbidden substructures. Open questions include whether the dichotomy can be pushed further to trigraphs whose underlying graphs have bounded treewidth, or whether a full classification for all trigraphs is attainable. Overall, the work bridges a gap between the well‑understood digraph world and the more intricate landscape of trigraph homomorphisms, offering both theoretical insight and a concrete algorithmic toolkit for a substantial subclass of problems.