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Min-Max orderings correspond to conservative lattice polymorphisms. Digraphswith Min-Max orderings have polynomial time solvable minimum cost homomorphismproblems. They can also be viewed as digraph analogues of proper intervalgraphs and bigraphs.  We give a forbidden structure characterization of digraphs with a Min-Maxordering which implies a polynomial time recognition algorithm. We alsosimilarly characterize digraphs with an extended Min-Max ordering, and we applythis characterization to prove a conjectured form of dichotomy for minimum costhomomorphism problems.

Duality for Min-Max Orderings and Dichotomy for Min Cost Homomorphisms