For digraphs G and H, a homomorphism of G to H is a mapping $f:\
V(G)\dom V(H)suchthatuv\in A(G)impliesf(u)f(v)\in A(H).Ifmoreovereachvertexu \in V(G)isassociatedwithcostsc_i(u), i \in V(H),thenthecostofahomomorphismfis\sum_{u\in V(G)}c_{f(u)}(u).ForeachfixeddigraphH, the {\em minimum cost homomorphism problem} for H,denotedMinHOM(H),isthefollowingproblem.GivenaninputdigraphG,togetherwithcostsc_i(u),u\in V(G),i\in V(H),andanintegerk,decideifGadmitsahomomorphismtoHofcostnotexceedingk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs H(everyvertexofHhasaloop).ItisknownthattheproblemMinHOM(H)ispolynomialtimesolvableifthedigraphH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <sothati<j, s<randir, js
\in A(H)implythatis \in A(H)andjr \in A(H).WegiveaforbiddeninducedsubgraphcharacterizationofreflexivedigraphswithaMinβMaxordering;ourcharacterizationimpliesapolynomialtimetestfortheexistenceofaMinβMaxordering.Usingthischaracterization,weshowthatforareflexivedigraphH$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs