Countable Graphs Research Articles

Permutation monoids and MB-homogeneity for graphs and relational structures

In this paper we investigate the connection between infinite permutation monoids and bimorphism monoids of first-order structures. Taking our lead from the study of automorphism groups of structures as infinite permutation groups and the more recent developments in the field of homomorphism-homogeneous structures, we establish a series of results that underline this connection. Of particular interest is the idea of MB-homogeneity; a relational structure M is MB-homogeneous if every monomorphism between finite substructures of M extends to a bimorphism of M. The results in question include a characterisation of closed permutation monoids, a Fraïssé-like theorem for MB-homogeneous structures, and the construction of 2ℵ0 pairwise non-isomorphic countable MB-homogeneous graphs. We prove that any finite group arises as the automorphism group of some MB-homogeneous graph and use this to construct oligomorphic permutation monoids with any given finite group of units. We also consider MB-homogeneity for various well-known examples of homogeneous structures and in particular give a complete classification of countable homogeneous undirected graphs that are also MB-homogeneous.

Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

For $n\geq 3$, let $(H_n, E)$ denote the $n$th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on $n$ vertices. We show that for all structures $\Gamma$ with domain $H_n$ whose relations are first-order definable in $(H_n,E)$ the constraint satisfaction problem for $\Gamma$ either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.

Which subsets of an infinite random graph look random?

AbstractGiven a countable graph, we say a set A of its vertices is universal if it contains every countable graph as an induced subgraph, and A is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on , (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on will almost surely contain non‐universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.

Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in [0,1/2].

THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM

AbstractWe study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

On isometry and isometric embeddability between ultrametric Polish spaces

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set D of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of D. When D contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If D is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length ω1 which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least ω+3.We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.

A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

Universal locally finite maximally homogeneous semigroups and inverse semigroups

Abstract In 1959, Philip Hall introduced the locally finite group 𝒰 {\mathcal{U}} , today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in 𝒰 {\mathcal{U}} . It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraïssé limit of the class of all finite groups. Since its introduction Hall’s group and several natural generalisations have been studied widely. In this article we use a generalisation of Fraïssé’s theory to construct a countable, universal, locally finite semigroup 𝒯 {\mathcal{T}} , that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup ℐ {\mathcal{I}} which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups 𝒯 {\mathcal{T}} and ℐ {\mathcal{I}} are the natural counterparts of Hall’s universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall’s group itself.

Copies of the random graph

Let 〈R,∼〉 be the Rado graph, Emb(R) the monoid of its self-embeddings, P(R)={f(R):f∈Emb(R)} the set of copies of R contained in R, and IR the ideal of subsets of R which do not contain a copy of R. We consider the poset 〈P(R),⊂〉, the algebra P(R)/IR, and the inverse of the right Green's preorder on Emb(R), and show that these preorders are forcing equivalent to a two step iteration of the form P⁎π, where the poset P is similar to the Sacks perfect set forcing: adds a generic real, has the ℵ0-covering property and, hence, preserves ω1, has the Sacks property and does not produce splitting reals, while π codes an ω-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph.

Breaking graph symmetries by edge colourings

The distinguishing index D′(G) of a graph G is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pilśniak conjectured that if every non-trivial automorphism of a countable graph G moves infinitely many edges, then D′(G)≤2. We prove this conjecture.

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l1. Further we show that, for each ordinal α < ω1, the space C([0, ωα]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ωα+1.

A rigid Urysohn-like metric space

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of R R extends to an automorphism of R R . We construct a graph of the smallest uncountable cardinality ω 1 \omega _1 which has the same extension property as R R , yet its group of automorphisms is trivial. We also present a similar, although technically more complicated, construction of a complete metric space of density ω 1 \omega _1 , having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.

Reducts of the Henson graphs with a constant

Let (Hn,E) denote the Henson graph, the unique countable homogeneous graph whose age consists of all finite Kn-free graphs. In this note the reducts of the Henson graphs with a constant are determined up to first-order interdefinability. It is shown that up to first-order interdefinability (H3,E,0) has 13 reducts and (Hn,E,0) has 16 reducts for n≥4.

Strong Spatial Mixing in Homomorphism Spaces

Given a countable graph $\mathcal{G}$ and a finite graph $H$, we consider $Hom(\mathcal{G}, H)$ the set of graph homomorphisms from $\mathcal{G}$ to $H$ and we study Gibbs measures supported on $Hom(\mathcal{G}, H)$. We develop some sufficient and other necessary conditions for the existence of Gibbs specifications on $Hom(\mathcal{G}, H)$ satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph $H$ has a combinatorial property called dismantlability iff for every $\mathcal{G}$ of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that such Gibbs specification can be chosen to have weak spatial mixing. In addition, we exhibit a subfamily of graphs $H$ for which there exists Gibbs specifications satisfying strong spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs specification has strong spatial mixing.

Distinguishing Cartesian products of countable graphs

Distinguishing Cartesian products of countable graphs

Several notions of rank-width for countable graphs

We define several notions of rank-width for countable graphs. We compare, for each of them the width of a countable graph with the least upper bound of the widths of its finite induced subgraphs. A width measure has the compactness property if these two values are equal. One of our notions of rank-width for countable graphs uses quasi-trees (generalized trees where “paths” may have any order type) and has this property. So has linear rank-width, based on arbitrary linear orders. A more natural notion of rank-width based on countable cubic trees (we call it discrete rank-width) has a weaker type of compactness: the corresponding width is at most twice the least upper bound of the widths of the finite induced subgraphs. The notion of discrete linear rank-width, based on discrete linear orders has no compactness property.

Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

Abstract A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random graph, infinite disjoint unions of complete graphs K n {K_{n}} with n ∈ ℕ {n\in\mathbb{N}} vertices, the K n {K_{n}} -free graphs, finite unions of the infinite complete graph K ω {K_{\omega}} , and duals of such graphs. The groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} of automorphisms of such graphs Γ have a natural topology, which is compatible with multiplication and inversion, i.e. the groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} where Γ is ultrahomogeneous. We show that if Γ is ultrahomogeneous, then Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} has 2-generated dense subgroups, and that under certain conditions given f ∈ Aut ⁡ ( Γ ) {f\in\operatorname{Aut}(\Gamma)} there exists g ∈ Aut ⁡ ( Γ ) {g\in\operatorname{Aut}(\Gamma)} such that the subgroup generated by f and g is dense. We also show that, roughly speaking, g can be chosen with a high degree of freedom. For example, if Γ is either an infinite disjoint union of K n {K_{n}} or a finite union of K ω {K_{\omega}} , then g can be chosen to have any given finite set of orbit representatives.

Diagonality of actions and KMS weights

The paper contains a description of a connection between diagonal actions and certain KMS weights on groupoid $C^{*}$-algebras. It furthermore contains the realization of a graph $C^{*}$-algebra of a countable graph as the groupoid $C^{*}$-algebra of a local homeomorphism and applies the theory obtained on these algebras.

The distinguishing index of the Cartesian product of countable graphs

The distinguishing index D ′ ( G ) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is preserved only by the trivial automorphism. We derive some bounds for this parameter for infinite graphs. In particular, we investigate the distinguishing index of the Cartesian product of countable graphs. Finally, we prove that D ʹ( K 2 ℵ 0 ) = 2 , where K 2 ℵ 0 is the infinite dimensional hypercube.

Exchangeable graph-valued Feller processes

The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $$\sigma $$ -finite measure on the space of $$\{0,1\}\times \{0,1\}$$ -valued arrays. In discrete time, this characterization gives rise to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous time, the behavior is enriched by a Levy–Ito–Khintchine-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Every such process almost surely projects to a Feller process in the space of graph limits.