Minimum Cost Homomorphism Dichotomy for Locally In-Semicomplete Digraphs

Abstract

For digraphs Gand H, a homomorphism of Gto His a mapping such that uv? A(G) implies f(u)f(v) ? A(H). In the minimum cost homomorphism problemwe associate costs c i (u), u? V(G), i? V(H) with the mapping of uto iand the cost of a homomorphism fis defined ? u? V(G) c f(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 Gto Hand 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 His 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.