For digraphs G and H, a homomorphism of G to H is a mapping $f:\
V(G)\dom V(H)suchthatuv\in A(G)impliesf(u)f(v)\in A(H).If,moreover,eachvertexu \in V(G)isassociatedwithcostsc_i(u), i \in V(H),thenthecostofahomomorphismfis\sum_{u\in V(G)}c_{f(u)}(u).ForeachfixeddigraphH,theminimumcosthomomorphismproblemforH,denotedMinHOM(H),canbeformulatedasfollows:GivenaninputdigraphG,togetherwithcostsc_i(u),u\in V(G),i\in V(H),decidewhetherthereexistsahomomorphismofGtoH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
For digraphs G and H, a homomorphism of G to H is a mapping $f:\
V(G)\dom V(H)suchthatuv\in A(G)impliesf(u)f(v)\in A(H).Ifmoreovereachvertexu \in V(G)isassociatedwithcostsc_i(u), i \in V(H),thenthecostofahomomorphismfis\sum_{u\in V(G)}c_{f(u)}(u).ForeachfixeddigraphH, the {\em minimum cost homomorphism problem} for H,denotedMinHOM(H),isthefollowingproblem.GivenaninputdigraphG,togetherwithcostsc_i(u),u\in V(G),i\in V(H),andanintegerk,decideifGadmitsahomomorphismtoHofcostnotexceedingk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs H(everyvertexofHhasaloop).ItisknownthattheproblemMinHOM(H)ispolynomialtimesolvableifthedigraphH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <sothati<j, s<randir, js
\in A(H)implythatis \in A(H)andjr \in A(H).WegiveaforbiddeninducedsubgraphcharacterizationofreflexivedigraphswithaMin−Maxordering;ourcharacterizationimpliesapolynomialtimetestfortheexistenceofaMin−Maxordering.Usingthischaracterization,weshowthatforareflexivedigraphH$ 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
The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique‐ and rank‐width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique‐ and rank‐width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub‐depth, and with respect to other notions, such as well‐quasi‐ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types