Minimum Cost Homomorphism Dichotomy for Locally In-Semicomplete Digraphs

For digraphs G and H , a homomorphism of G to H is a mapping f : V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). In the minimum cost homomorphism problem we associate costs ci(u), u ∈ V (G), i ∈ V (H) with the mapping of u to i and the cost of a homomorphism f is defined ∑ u∈V (G) cf(u)(u) accordingly. Here the minimum cost homomorphism problem for a fixed digraph H , denoted by MinHOM(H), is to check whether there exists a homomorphism of G to H and to obtain one of minimum cost, if one does exit. The minimum cost homomorphism problem is now well understood for digraphs with loops. For loopless digraphs only partial results are known. In this paper, we find a full dichotomy classification of MinHom(H), when H is a locally in-semicomplete digraph. This is one of the largest classes of loopless digraphs for which such dichotomy classification has been proved. This paper extends the previous result for locally semicomplete digraphs.