Near-Unanimity Functions and Varieties of Reflexive Graphs

Abstract

Abstract. 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(x 1 , x 2 , . . . , x k ) is adjacent to g(x 1 , x 2 , . . . , x k ) whenever x i x i ∈ E(H) for each i = 1, 2, . . . , k, and g(x 1 , x 2 , . . . , 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 − √ 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.

Colouring, constraint satisfaction, and complexity

“…Bi-arc graphs are shown to be precisely the graphs with a near-unanimity polymorphism in [22], and have recently been shown also precisely the graphs with a Taylor operation [48]. (The latter fact confirms Conjecture 3.5 for conservative structures that are graphs.…”

“…Near-unanimity polymorphisms are fairly common [134,22]. For instance, if H is a digraph (structure with one binary relation) whose underlying undirected graph is a path, then H admits a simple majority polymorphism -namely f (u, v, w) being the middle of the vertices u, v, w on the path.…”

Bounded width problems and algebras

“…We note however that bounded strict width of structures is never preserved by the Feder-Vardi construction. The problem CSP(B) has strict width l if, for any partial map f from a structure I to B that does not extend to a full homomorphism, there exists a subset of its domain with at most l elements such that the restriction of f to this subset still does not extend ( [17] page 82, see also [4]). No matter what B is, its Feder-Vardi poset P is always a ramified poset, that is, P is a connected poset with at least two elements and has no element with a unique lower or upper cover.…”

Graphs Admitting $k$-NU Operations. Part 1: The Reflexive Case

“…Relational structures invariant under near-unanimity operations possess remarkable properties; algebras and graphs admitting NU operations, and especially the better known case of 3-ary or majority operations, have been studied extensively ( [1], [2], [3], [4], [5], [9], [10], [13], [21], [24], [25], [29], [33]). …”

“…Although some work was done in trying to generalise the classification results for majority strutures to those admitting NU operation of higher arity (see for instance [5], [10], [29], [31] and [32]), for a while the problem seemed quite hopeless. One major obstacle is that the metric point of view used to prove the results in the majority case seems impossible to adapt for arities 4 and up, and a new approach seemed necessary.…”