Near-Unanimity Functions and Varieties of Reflexive Graphs

Let $H$ be a graph and $k \geq 3$. A near-unanimity function of arity $k$ is a mapping $g$ from the $k$-tuples over $V(H)$ to $V(H)$ such that $g(x_1, x_2, \dots, x_k)$ is adjacent to $g(x'_1, x'_2, \dots, x'_k)$ whenever $x_i x'_i \in E(H)$ for each $i = 1, 2, \dots, k$, and $g(x_1, x_2, \dots, x_k) = a$ whenever at least $k-1$ of the $x_i$'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 $n -\sqrt{n}+1$, 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 $\neq$ 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 $\neq$ NP.