Near-Unanimity Functions and Varieties of Reflexive Graphs

Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) to V (H) such that g(x1, x2, . . . , xk) is adjacent to g(x ′ 1, x ′ 2, . . . , x ′ k) whenever xix ′ i ∈ E(H) for each i = 1, 2, . . . , k, and g(x1, x2, . . . , xk) = a whenever at least k − 1 of the xi’s equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity function, then the homomorphism extension (or retraction) problem for H is polynomial time solvable. We focus on near-unanimity functions on reflexive graphs. The best understood are reflexive chordal graphs H: they always admit a near-unanimity function. We bound the arity of these functions in several ways related to the size of the largest clique and the leafage of H, and we show that these bounds are tight. In particular, it will follow that the arity is bounded by n− √ n+1, where n = |V (H)|. We investigate substructures forbidden for reflexive graphs that admit a near-unanimity function. It will follow, for instance, that no reflexive cycle of length at least four admits a near-unanimity function of any arity. However, we exhibit nonchordal graphs which do admit near-unanimity functions. Finally, we characterize graphs which admit a conservative near-unanimity function. This characterization has been predicted by the results of Feder, Hell, and Huang. Specifically, those results imply that, if P = NP, the graphs with conservative near-unanimity functions are precisely the so-called bi-arc graphs. We give a proof of this statement without assuming P = NP.