Convex circuit-free coloration of an oriented graph
European Journal of Combinatorics
Constraint Satisfaction Problems with Infinite Templates
Complexity of Constraints
Paths, cycles and circular colorings in digraphs
Theoretical Computer Science
Gallai's Theorem for List Coloring of Digraphs
SIAM Journal on Discrete Mathematics
Survey: Colouring, constraint satisfaction, and complexity
Computer Science Review
Finding small separators in linear time via treewidth reduction
ACM Transactions on Algorithms (TALG)
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An acyclic homomorphism of a digraph $D$ into a digraph $F$ is a mapping $\phi\colon V(D) \to V(F)$ such that for every arc $uv\in E(D)$, either $\phi(u)=\phi(v)$ or $\phi(u)\phi(v)$ is an arc of $F$, and for every vertex $v\in V(F)$, the subgraph of $D$ induced on $\phi^{-1}(v)$ is acyclic. For each fixed digraph $F$ we consider the following decision problem: Does a given input digraph $D$ admit an acyclic homomorphism to $F$? We prove that this problem is NP-complete unless $F$ is acyclic, in which case it is polynomial time solvable. From this we conclude that it is NP-complete to decide if the circular chromatic number of a given digraph is at most $q$, for any rational number $q 1$. We discuss the complexity of the problems restricted to planar graphs. We also refine the proof to deduce that certain $F$-coloring problems are NP-complete.