The complexity of frugal digraph homomorphisms
For graph theorists and complexity theorists, this provides a complete complexity classification for a family of homomorphism problems with bounded multiplicity.
The paper establishes a structural dichotomy theorem for t-frugal homomorphisms of digraphs when t ≥ 2, showing that the problem is either polynomial-time solvable or NP-complete based on the structure of the target digraph H.
For an integer $t \geq 1$, a homomorphism of a digraph G to a digraph $H$ is $t$-frugal if no more than $t$ in-neighbours of any vertex of $G$ have the same image. There is a dichotomy theorem based on structural properties when $t=1$ and $H$ has a loop at each vertex, but there is unlikely to be such a theorem for general digraphs $H$. For $t \geq 2$ we describe a structural dichotomy theorem for $t$-frugal homomorphisms of general digraphs.