Semilattice polymorphisms and chordal graphs

We investigate the class of reflexive graphs that admit semilattice polymorphisms, and in doing so, give an algebraic characterisation of chordal graphs. In particular, we show that a graph G is chordal if and only if it has a semilattice polymorphism such that G is a subgraph of the comparability graph of the semilattice. Further, we find a new characterisation of the leafage of a chordal graph in terms of the width of the semilattice polymorphisms it admits. Finally, we introduce obstructions to various types of semilattice polymorphisms, and in doing so, show that the class of reflexive graphs admitting semilattice polymorphisms is not a variety. These are, to our knowledge, the first structural results on graphs with semilattice polymorphisms, beyond the conservative realm.